حمّل تطبيق منهاجي الجديد

منهاجي صار أسرع من خلال التطبيق

  أتدرب وأحل المسائل

أتدرب وأحل المسائل

المعادلات التفاضلية

أحدد إذا كان الاقتران المعطى حلاً للمعادلة التفاضلية في كل مما يأتي:

begin mathsize 20px style y equals square root of x semicolon x y to the power of straight prime minus y equals 0 end style (1)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y to the power of straight prime equals fraction numerator 1 over denominator 2 square root of x end fraction end cell row blank cell x y to the power of straight prime minus y equals x fraction numerator 1 over denominator 2 square root of x end fraction minus square root of x equals 1 half square root of x minus square root of x equals negative 1 half square root of x not equal to 0 end cell end table end style

إذن، begin mathsize 20px style y equals square root of x end style ليس حلاً للمعادلة التفاضلية begin mathsize 20px style x y to the power of straight prime minus y equals 0 end style

begin mathsize 20px style y equals x ln invisible function application x minus 5 x plus 7 semicolon y to the power of ′′ minus 1 over x equals 0 end s (2)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y to the power of straight prime equals x left parenthesis 1 over x right parenthesis plus ln invisible function application x minus 5 equals ln invisible function application x minus 4 end cell row blank cell y to the power of ′′ equals 1 over x end cell row blank cell y to the power of ′′ minus 1 over x equals 1 over x minus 1 over x equals 0 end cell end table e

إذن، begin mathsize 20px style y equals x ln invisible function application x minus 5 x plus 7 end style هو حل للمعادلة التفاضلية begin mathsize 20px style y to the power of ′′ minus 1 over x equals 0 end s

begin mathsize 20px style y equals tan invisible function application x semicolon y to the power of straight prime plus y squared equals 1 end style (3)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y to the power of straight prime equals sec squared invisible function application x end cell row blank cell y to the power of straight prime plus y squared equals sec squared invisible function application x plus tan squared invisible function application x equals 1 plus 2 tan squared invisible function application x not equal to 1 end cell end table end style

إذن، begin mathsize 20px style y equals tan invisible function application x end style ليس حلاً للمعادلة التفاضلية begin mathsize 20px style y to the power of straight prime plus y squared equals 1 end style

begin mathsize 20px style y equals e to the power of x plus 3 x e to the power of x semicolon y to the power of ′′ minus 2 y to the power of straight prime plus y equals 0 end s (4)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y to the power of straight prime equals e to the power of x plus 3 x e to the power of x plus 3 e to the power of x equals 4 e to the power of x plus 3 x e to the power of x end cell row blank cell y to the power of ′′ equals 4 e to the power of x plus 3 x e to the power of x plus 3 e to the power of x equals 7 e to the power of x plus 3 x e to the power of x end cell row blank cell y to the power of ′′ minus 2 y to the power of straight prime plus y equals 7 e to the power of x plus 3 x e to the power of x minus 8 e to the power of x minus 6 x e to the power of x plus e to the power of x plus 3 x e to the power of x equals 0 end cell end table e

إذن، begin mathsize 20px style y equals e to the power of x plus 3 x e to the power of x end style هو حل للمعادلة التفاضلية begin mathsize 20px style y to the power of ′′ minus 2 y to the power of straight prime plus y equals 0 end s

أحل كلاً من المعادلات التفاضلية الآتية:

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 3 x square root of y end style (5)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator d y over denominator d x end fraction equals 3 x square root of y end cell blank row cell fraction numerator d y over denominator square root of y end fraction equals 3 x d x end cell cell not stretchy rightwards double arrow integral fraction numerator d y over denominator square root of y end fraction equals integral 3 x d x end cell row blank cell not stretchy rightwards double arrow 2 y to the power of 1 half end exponent equals 3 over 2 x squared plus C end cell end table end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction plus fraction numerator 3 x over denominator y squared end fraction equals 0 end style (6)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals negative fraction numerator 3 x over denominator y squared end fraction not stretchy rightwards double arrow y squared d y equals negative 3 x d x end cell row blank cell integral y squared d y equals integral negative 3 x d x end cell row blank cell not stretchy rightwards double arrow 1 third y cubed equals negative 3 over 2 x squared plus C end cell end table end style

  begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals cos invisible function application x sin invisible function application y end style (7)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals c o s invisible function application x s i n invisible function application y d x end cell row blank cell fraction numerator d y over denominator s i n invisible function application y end fraction equals c o s invisible function application x d x end cell row blank cell not stretchy rightwards double arrow integral c s c invisible function application y d y equals integral c o s invisible function application x d x end cell end table
table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral c s c invisible function application y d y equals integral c s c invisible function application y cross times fraction numerator c s c invisible function application y plus c o t invisible function application y over denominator c s c invisible function application y plus c o t invisible function application y end fraction d y end cell row blank cell space of 1em equals integral fraction numerator c s c squared invisible function application y plus c s c invisible function application y c o t invisible function application y over denominator c s c invisible function application y plus c o t invisible function application y end fraction d y end cell row blank cell space of 1em equals negative integral fraction numerator negative left parenthesis c s c squared invisible function application y plus c s c invisible function application y c o t invisible function application y right parenthesis over denominator c s c invisible function application y plus c o t invisible function application y end fraction d y equals negative l n invisible function application vertical line c s c invisible function application y plus c o t invisible function application y vertical line end cell row blank cell not stretchy rightwards double arrow negative l n invisible function application vertical line c s c invisible function application y plus c o t invisible function application y vertical line equals s i n invisible function application x plus C end cell end table end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals x over left parenthesis x squared plus 1 right parenthesis squared end style (8)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell d y equals x over left parenthesis x squared plus 1 right parenthesis squared d x end cell row blank cell integral d y equals integral x over left parenthesis x squared plus 1 right parenthesis squared d x end cell end table
table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals x squared plus 1 not stretchy rightwards double arrow d x equals fraction numerator d u over denominator 2 x end fraction end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow integral x over left parenthesis x squared plus 1 right parenthesis squared d x equals integral x over u squared fraction numerator d u over denominator 2 x end fraction end cell cell equals 1 half integral 1 over u squared d u end cell row blank cell equals negative fraction numerator 1 over denominator 2 u end fraction plus C equals negative fraction numerator 1 over denominator 2 left parenthesis x squared plus 1 right parenthesis end fraction plus C end cell end table end cell row blank cell not stretchy rightwards double arrow integral d y equals integral x over left parenthesis x squared plus 1 right parenthesis squared d x not stretchy rightwards double arrow y equals negative fraction numerator 1 over denominator 2 left parenthesis x squared plus 1 right parenthesis end fraction plus C end cell end table end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals x e to the power of x plus y end exponent end style (9)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals x e to the power of x e to the power of y not stretchy rightwards double arrow fraction numerator d y over denominator e to the power of y end fraction equals x e to the power of x d x end cell row blank cell integral fraction numerator d y over denominator e to the power of y end fraction equals integral x e to the power of x d x end cell row blank cell not stretchy rightwards double arrow integral e to the power of negative y end exponent d y equals integral x e to the power of x d x end cell row blank cell not stretchy rightwards double arrow negative e to the power of negative y end exponent equals integral x e to the power of x d x space open parentheses الأجزاء space نستخدم close parentheses end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals x space of 1em d v equals e to the power of x d x end cell row blank cell d u equals d x space of 1em v equals e to the power of x end cell row blank cell not stretchy rightwards double arrow integral x e to the power of x d x equals x e to the power of x minus integral e to the power of x d x equals x e to the power of x minus e to the power of x plus C end cell row blank cell not stretchy rightwards double arrow negative e to the power of negative y end exponent equals x e to the power of x minus e to the power of x plus C end cell end ta

begin mathsize 20px style e to the power of negative 1 divided by x end exponent fraction numerator d y over denominator d x end fraction equals x to the power of negative 2 end exponent y squared end style (10)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator y squared end fraction equals x to the power of negative 2 end exponent over e to the power of negative 1 over x end exponent d x equals e to the power of 1 over x end exponent over x squared d x end cell row blank cell integral y to the power of negative 2 end exponent d y equals integral e to the power of 1 over x end exponent over x squared d x end cell row blank cell not stretchy rightwards double arrow negative y to the power of negative 1 end exponent equals integral e to the power of 1 over x end exponent over x squared d x space open parentheses التعويض space نستخدم close parentheses end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals 1 over x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals negative 1 over x squared not stretchy rightwards double arrow d x equals negative x squared d u end cell row blank cell not stretchy rightwards double arrow integral e to the power of 1 over x end exponent over x squared d x equals integral e to the power of u over x squared cross times negative x squared d u equals integral negative e to the power of u d u equals negative e to the power of u plus C equals negative e to the power of 1 over x end exponent plus C end cell row blank cell negative y to the power of negative 1 end exponent equals integral e to the power of 1 over x end exponent over x squared d x not stretchy rightwards double arrow 1 over y equals e to the power of 1 over x end exponent plus C end cell end ta

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator x y over denominator x minus 3 end fraction end style (11)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator y end fraction equals fraction numerator x over denominator x minus 3 end fraction d x end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral fraction numerator x over denominator x minus 3 end fraction d x end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral left parenthesis 1 plus fraction numerator 3 over denominator x minus 3 end fraction right parenthesis d x end cell row blank cell ln invisible function application vertical line y vertical line equals x plus 3 ln invisible function application vertical line x minus 3 vertical line plus C end cell end table end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator 3 x squared sin squared invisible function application y over denominator x cubed plus 2 end fraction end style (12)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator sin squared invisible function application y end fraction equals fraction numerator 3 x squared over denominator left parenthesis x cubed plus 2 right parenthesis end fraction d x end cell row blank cell integral fraction numerator d y over denominator sin squared invisible function application y end fraction equals integral fraction numerator 3 x squared over denominator left parenthesis x cubed plus 2 right parenthesis end fraction d x end cell row blank cell integral csc squared invisible function application y d y equals integral fraction numerator 3 x squared over denominator x cubed plus 2 end fraction d x end cell row blank cell negative cot invisible function application y equals ln invisible function application vertical line x cubed plus 2 vertical line plus C end cell end table end style

 begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals y cubed ln invisible function application x end style (13)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell dy over straight y cubed equals ln invisible function application xdx end cell row blank cell integral dy over straight y cubed equals integral ln invisible function application xdx space open parentheses الأجزاء space نستخدم close parentheses end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell straight u equals ln invisible function application straight x space of 1em dv equals dx end cell row blank cell du equals dx over straight x space of 1em straight v equals straight x end cell row blank cell integral ln invisible function application xdx equals xln invisible function application straight x minus integral straight x dx over straight x equals xln invisible function application straight x minus straight x plus straight C end cell row blank cell not stretchy rightwards double arrow integral straight y to the power of negative 3 end exponent dy equals integral ln invisible function application xdx not stretchy rightwards double arrow negative 1 half straight y to the power of negative 2 end exponent equals xln invisible function application straight x minus straight x plus straight C end cell end ta

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 2 x cubed left parenthesis y squared minus 1 right parenthesis end style (14)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator y squared minus 1 end fraction equals 2 x cubed d x end cell row blank cell integral fraction numerator d y over denominator y squared minus 1 end fraction equals integral 2 x cubed d x space open parentheses الجزئية space الكسور space نستخدم close parentheses
nd cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 1 over denominator y squared minus 1 end fraction equals fraction numerator 1 over denominator left parenthesis y minus 1 right parenthesis left parenthesis y plus 1 right parenthesis end fraction equals fraction numerator A over denominator y minus 1 end fraction plus fraction numerator B over denominator y plus 1 end fraction end cell row blank cell A left parenthesis y plus 1 right parenthesis plus B left parenthesis y minus 1 right parenthesis equals 1 end cell row blank cell y equals 1 not stretchy rightwards double arrow A equals 1 half end cell row blank cell y equals negative 1 not stretchy rightwards double arrow B equals negative 1 half end cell row blank cell not stretchy rightwards double arrow fraction numerator 1 over denominator y squared minus 1 end fraction equals fraction numerator 1 half over denominator y minus 1 end fraction plus fraction numerator negative 1 half over denominator y plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow integral fraction numerator d y over denominator y squared minus 1 end fraction equals integral 2 x cubed d x not stretchy rightwards double arrow integral left parenthesis fraction numerator 1 half over denominator y minus 1 end fraction plus fraction numerator negative 1 half over denominator y plus 1 end fraction right parenthesis d y equals integral 2 x cubed d x end cell row blank cell not stretchy rightwards double arrow 1 half l n invisible function application vertical line y minus 1 vertical line minus 1 half l n invisible function application vertical line y plus 1 vertical line equals 1 half x to the power of 4 plus C end cell

begin mathsize 20px style y fraction numerator d y over denominator d x end fraction equals sin cubed invisible function application x cos squared invisible function application x end style (15)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell y d y equals s i n cubed invisible function application x c o s squared invisible function application x d x end cell row blank cell integral y d y equals integral s i n cubed invisible function application x c o s squared invisible function application x d x space open parentheses التعويض space نستخدم close parentheses end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals c o s invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator negative s i n invisible function application x end fraction end cell row blank cell integral s i n cubed invisible function application x c o s squared invisible function application x d x equals integral s i n cubed invisible function application x u squared fraction numerator d u over denominator negative s i n invisible function application x end fraction end cell row blank cell equals integral negative s i n squared invisible function application x u squared d u equals integral left parenthesis negative 1 plus c o s squared invisible function application x right parenthesis u squared d u end cell row blank cell equals integral left parenthesis negative 1 plus u squared right parenthesis u squared d u equals integral left parenthesis u to the power of 4 minus u squared right parenthesis d u end cell row blank cell equals 1 fifth u to the power of 5 minus 1 third u cubed plus C equals 1 fifth c o s to the power of 5 invisible function application x minus 1 third c o s cubed invisible function application x plus C end cell row blank cell not stretchy rightwards double arrow integral y d y equals integral s i n cubed invisible function application x c o s squared invisible function application x d x end cell row blank cell not stretchy rightwards double arrow 1 half y squared equals 1 fifth c o s to the power of 5 invisible function application x minus 1 third c o s cubed invisible function application x plus C end cell end ta

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals square root of x y end root end style (16)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator square root of y end fraction equals square root of x d x end cell row blank cell integral fraction numerator d y over denominator square root of y end fraction equals integral square root of x d x end cell row blank cell integral y to the power of negative 1 half end exponent d y equals integral x to the power of 1 half end exponent d x end cell row blank cell 2 y to the power of 1 half end exponent equals 2 over 3 x to the power of 3 over 2 end exponent plus C end cell end table end style

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals y ln invisible function application square root of x end style (17)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator y end fraction equals l n invisible function application x to the power of 1 half end exponent d x end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral l n invisible function application x to the power of 1 half end exponent d x end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral 1 half l n invisible function application x d x space open parentheses الأجزاء space نستخدم close parentheses end c
l end table\ntable attributes columnalign left right columnspacing 1em end attributes row cell u equals l n invisible function application x end cell cell d v equals d x end cell row cell d u equals fraction numerator d x over denominator x end fraction end cell cell v equals x end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral l n invisible function application x d x equals x l n invisible function application x minus integral x fraction numerator d x over denominator x end fraction equals x l n invisible function application x minus x plus C end cell row blank cell not stretchy rightwards double arrow integral 1 half l n invisible function application x d x equals 1 half x l n invisible function application x minus 1 half x plus C end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral 1 half l n invisible function application x d x not stretchy rightwards double arrow l n invisible function application vertical line y vertical line equals 1 half x l n invisible function application x minus 1 half x plus C end cell end ta

begin mathsize 20px style left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis fraction numerator d y over denominator d x end fraction equals negative 3 left parenthesis y minus 2 right parenthesis end style (18)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis d y equals negative 3 left parenthesis y minus 2 right parenthesis d x end cell row blank cell integral negative 1 third fraction numerator d y over denominator y minus 2 end fraction equals integral fraction numerator d x over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction space open parentheses الجزئية space الكسور space نستخدم close parentheses
nd cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 1 over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction equals fraction numerator A over denominator 2 x plus 1 end fraction plus fraction numerator B over denominator x plus 2 end fraction end cell row blank cell A left parenthesis x plus 2 right parenthesis plus B left parenthesis 2 x plus 1 right parenthesis equals 1 end cell row blank cell x equals negative 1 half not stretchy rightwards double arrow A equals 2 over 3 end cell row blank cell x equals negative 2 not stretchy rightwards double arrow B equals negative 1 third end cell row blank cell not stretchy rightwards double arrow fraction numerator 1 over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction equals fraction numerator 2 over 3 over denominator 2 x plus 1 end fraction plus fraction numerator negative 1 third over denominator x plus 2 end fraction end cell row blank cell not stretchy rightwards double arrow integral negative 1 third fraction numerator d y over denominator y minus 2 end fraction equals integral fraction numerator d x over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end fraction end cell row blank cell not stretchy rightwards double arrow negative 1 third l n invisible function application vertical line y minus 2 vertical line equals 1 third l n invisible function application vertical line 2 x plus 1 vertical line minus 1 third l n invisible function application vertical line x plus 2 vertical line plus C end cell row blank cell not stretchy rightwards double arrow negative l n invisible function application vertical line y minus 2 vertical line equals l n invisible function application vertical line 2 x plus 1 vertical line minus l n invisible function application vertical line x plus 2 vertical line plus C end cell row blank cell not stretchy rightwards double arrow negative l n invisible function application vertical line y minus 2 vertical line equals l n invisible function application vertical line fraction numerator 2 x plus 1 over denominator x plus 2 end fraction vertical line plus C end cell

أجد الحل الخاص الذي يحقق الشرط الأولي المعطى لكل من المعادلات التفاضلية الآتية:

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals y squared square root of 4 minus x end root semicolon y left parenthesis 1 right parenthesis equals 2 end style (19)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals y squared square root of 4 minus x end root end cell row blank cell fraction numerator d y over denominator y squared end fraction equals square root of 4 minus x end root d x end cell row blank cell integral fraction numerator d y over denominator y squared end fraction equals integral square root of 4 minus x end root d x end cell row blank cell integral y to the power of negative 2 end exponent d y equals integral left parenthesis 4 minus x right parenthesis to the power of 1 half end exponent d x end cell row blank cell negative 1 over y equals negative 2 over 3 left parenthesis 4 minus x right parenthesis to the power of 3 over 2 end exponent plus C space open parentheses العام space الحل close parentheses end cell end table

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell negative 1 half equals negative 2 square root of 3 plus C not stretchy rightwards double arrow C equals 2 square root of 3 minus 1 half space open parentheses left parenthesis 1 comma 2 right parenthesis space بتعويض close parentheses end cell row blank cell negative 1 over y equals negative 2 over 3 left parenthesis 4 minus x right parenthesis to the power of 3 over 2 end exponent plus 2 square root of 3 minus 1 half space open parentheses الخاص space الحل close parentheses end cell end

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator 2 sin squared invisible function application x over denominator y end fraction semicolon y left parenthesis 0 right parenthesis equals 1 end style (20)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals fraction numerator 2 sin squared invisible function application x over denominator y end fraction end cell row blank cell y d y equals 2 sin squared invisible function application x d x end cell row blank cell integral y d y equals integral 2 sin squared invisible function application x d x end cell row blank cell integral y d y equals integral left parenthesis 1 minus cos invisible function application 2 x right parenthesis d x end cell row blank cell 1 half y squared equals x minus 1 half sin invisible function application 2 x plus C space open parentheses العام space الحل close parentheses end cell end table

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 1 half equals 0 plus C not stretchy rightwards double arrow C equals 1 half space open parentheses open parentheses 0 comma 1 close parentheses space يتعويض close parentheses end cell row blank cell 1 half y squared equals x minus 1 half sin invisible function application 2 x plus 1 half space open parentheses الخاص space الحل close parentheses end cell end

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals 2 cos squared invisible function application x cos squared invisible function application y semicolon y left parenthesis 0 right parenthesis equals pi over 4 end style (21)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals 2 cos squared invisible function application x cos squared invisible function application y end cell row blank cell fraction numerator d y over denominator cos squared invisible function application y end fraction equals 2 cos squared invisible function application x d x end cell row blank cell integral fraction numerator d y over denominator cos squared invisible function application y end fraction equals integral 2 cos squared invisible function application x d x end cell row blank cell integral sec squared invisible function application y d y equals integral left parenthesis 1 plus cos invisible function application 2 x right parenthesis d x end cell row blank cell tan invisible function application y equals x plus 1 half sin invisible function application 2 x plus C space open parentheses العام space الحل close parentheses end cell end table

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 1 equals 0 plus 0 plus C space of 1em open parentheses left parenthesis 0 comma pi over 4 right parenthesis space بتعويض close parentheses end cell row blank cell C equals 1 end cell row blank cell tan invisible function application y equals x plus 1 half sin invisible function application 2 x plus 1 space open parentheses الخاص space الحل close parentheses end cell end

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator cos invisible function application x e to the power of sin invisible function application x end exponent over denominator e to the power of y end fraction semicolon y left parenthesis pi right parenthesis equals 0 end style (22)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals fraction numerator c o s invisible function application x e to the power of s i n invisible function application x end exponent over denominator e to the power of y end fraction end cell row blank cell integral e to the power of y d y equals integral c o s invisible function application x e to the power of s i n invisible function application x end exponent d x space open parentheses التعويض space نستخدم close parentheses end c
l end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell u equals s i n invisible function application x not stretchy rightwards double arrow fraction numerator d u over denominator d x end fraction equals c o s invisible function application x not stretchy rightwards double arrow d x equals fraction numerator d u over denominator c o s invisible function application x end fraction end cell row blank cell table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral c o s invisible function application x e to the power of s i n invisible function application x end exponent d x equals integral c o s invisible function application x e to the power of u cross times fraction numerator d u over denominator c o s invisible function application x end fraction equals integral e to the power of u d u equals e to the power of u plus C end cell row blank cell equals e to the power of s i n invisible function application x end exponent plus C end cell end table end cell row blank cell not stretchy rightwards double arrow integral e to the power of y d y equals integral c o s invisible function application x e to the power of s i n invisible function application x end exponent d x end cell row blank cell e to the power of y equals e to the power of s i n invisible function application x end exponent plus C space open parentheses العام space الحل close parenthe
s end cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell e to the power of 0 equals e to the power of 0 plus C space open parentheses open parentheses straight pi comma 0 close parentheses space بتعويض close parentheses end cell row blank cell not stretchy rightwards double arrow C equals 0 end cell row blank cell e to the power of y equals e to the power of s i n invisible function application x end exponent space open parentheses الخاص space الحل close par

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction semicolon y left parenthesis 3 right parenthesis equals 8 end style (23)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction end cell row blank cell integral d y equals integral fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction d x space open parentheses الجزئية space الكسور space نستخدم close parentheses
nd cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction equals fraction numerator A over denominator 3 x minus 8 end fraction plus fraction numerator B over denominator x minus 2 end fraction end cell row blank cell A left parenthesis x minus 2 right parenthesis plus B left parenthesis 3 x minus 8 right parenthesis equals 8 x minus 18 end cell row blank cell x equals 2 not stretchy rightwards double arrow B equals 1 end cell row blank cell x equals 8 over 3 not stretchy rightwards double arrow A equals 5 end cell row blank cell not stretchy rightwards double arrow fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction equals fraction numerator 5 over denominator 3 x minus 8 end fraction plus fraction numerator 1 over denominator x minus 2 end fraction end cell row blank cell not stretchy rightwards double arrow integral d y equals integral fraction numerator 8 x minus 18 over denominator left parenthesis 3 x minus 8 right parenthesis left parenthesis x minus 2 right parenthesis end fraction d x not stretchy rightwards double arrow y equals integral left parenthesis fraction numerator 5 over denominator 3 x minus 8 end fraction plus fraction numerator 1 over denominator x minus 2 end fraction right parenthesis d x end cell row blank cell not stretchy rightwards double arrow y equals 5 over 3 l n invisible function application vertical line 3 x minus 8 vertical line plus l n invisible function application vertical line x minus 2 vertical line plus C space open parentheses العام space الحل close pa
ntheses end cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 8 equals 0 plus 0 plus C not stretchy rightwards double arrow C equals 8 space open parentheses open parentheses 3 comma 8 close parentheses space بتعويض close parentheses end cell row blank cell y equals 5 over 3 l n invisible function application vertical line 3 x minus 8 vertical line plus l n invisible function application vertical line x minus 2 vertical line plus 8 space open parentheses الخاص space الحل clo

begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator x y end fraction semicolon y left parenthesis e right parenthesis equals 1 end style (24)

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator x y end fraction end cell row blank cell integral y d y equals integral fraction numerator d x over denominator x end fraction end cell row blank cell 1 half y squared equals l n invisible function application vertical line x vertical line plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 1 half equals 1 plus C space open parentheses open parentheses e comma 1 close parentheses space بتعويض close parentheses end cell row blank cell not stretchy rightwards double arrow C equals negative 1 half end
ell end table\n1 over z y to the power of z equals l n invisible function application vertical line x vertical line minus 1 half space open parentheses الخاص space الحل clo

(25) تتحرك سيارة في مسار مستقيم، ويعطى تسارعها بالمعادلة التفاضلية: begin mathsize 20px style fraction numerator d v over denominator d t end fraction equals 10 minus 0.5 v end style، حيث begin mathsize 20px style t end style الزمن بالثواني، وbegin mathsize 20px style v end style سـرعتها المتجهة بالمتر لكل ثانية، أجد السرعة المتجهة للسيارة بعد begin mathsize 20px style t end style ثانية من بدء حركتها، علماً بأن السيارة تحركت من وضع السكون.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d v over denominator d t end fraction equals 10 minus 0.5 v end cell row blank cell fraction numerator d v over denominator 10 minus 0.5 v end fraction equals d t end cell row blank cell integral fraction numerator d v over denominator 10 minus 0.5 v end fraction equals integral d t end cell row blank cell negative 2 l n invisible function application vertical line 10 minus 0.5 v vertical line equals t plus C not stretchy rightwards double arrow l n invisible function application vertical line 10 minus 0.5 v vertical line equals negative t over 2 plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell l n invisible function application 10 equals 0 plus C not stretchy rightwards double arrow C equals l n invisible function application 10 space open parentheses t equals 0 comma v equals 0 space بتعويض close parentheses end cell row blank cell not stretchy rightwards double arrow l n invisible function application vertical line 10 minus 0.5 v vertical line equals negative t over 2 plus l n invisible function application 10 not stretchy rightwards double arrow l n invisible function application vertical line fraction numerator 10 minus 0.5 v over denominator 10 end fraction vertical line equals negative t over 2 space open parentheses الخاص space الحل close parentheses end

إذن، يمكن نمذجة السرعة المتجهة للسيارة بعد t ثانية من بددء حركتها بالعلاقة الآتية:

begin mathsize 20px style ln invisible function application vertical line fraction numerator 10 minus 0.5 v over denominator 10 end fraction vertical line equals negative t over 2 end style

ذئاب(26) ذئاب: يمكن نمذجة معدل تغير عدد الذئاب في إحدى الغابات بالمعادلة التفاضلية: begin mathsize 20px style fraction numerator d N over denominator d t end fraction equals 260 minus 0.4 N end style، حيث begin mathsize 20px style N end style عدد الذئاب في الغابة بعد begin mathsize 20px style t end style سنة من بدء دراسة عليها. أجد عدد الذئاب في الغابة بعد 3 سنوات من بدء الدراسة، علماً بأن عددها عند بده الدراسة هو 300 ذئب.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d N over denominator d t end fraction equals 260 minus 0.4 N equals 0.4 left parenthesis 650 minus N right parenthesis end cell row blank cell fraction numerator d N over denominator 650 minus N end fraction equals 0.4 d t end cell row blank cell integral fraction numerator d N over denominator 650 minus N end fraction equals integral 0.4 d t end cell row blank cell negative I n invisible function application vertical line 650 minus N vertical line equals 0.4 t plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell negative l n invisible function application 350 equals 0 plus C not stretchy rightwards double arrow C equals negative l n invisible function application 350 space open parentheses t equals 0 comma straight N equals 300 space نعوض close parentheses end cell row blank cell not stretchy rightwards double arrow negative l n invisible function application vertical line 650 minus N vertical line equals 0.4 t minus l n invisible function application 350 end cell row blank cell not stretchy rightwards double arrow l n invisible function application vertical line fraction numerator 350 over denominator 650 minus N end fraction vertical line equals 0.4 t space open parentheses الخاص space الحل close parentheses end ce

لا يمكن أن يكون 650=N لأن 0 In غير معرف ولأن 300=N عندما 0=t والاقتران (N t متصل فلا يمكن أن يكون N أكبر من 650، ولذا فإن N>0 ويكون begin mathsize 20px style vertical line 650 minus N vertical line end style مسار لـ begin mathsize 20px style 650 minus N end style

begin mathsize 20px style not stretchy rightwards double arrow ln invisible function application vertical line fraction numerator 350 over denominator 650 minus N end fraction vertical line equals ln invisible function application left parenthesis fraction numerator 350 over denominator 650 minus N end fraction right parenthesis equals 0.4 t space open parentheses N space نجد space t equals 3 space نعوض close pare
heses\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell ln invisible function application left parenthesis fraction numerator 350 over denominator 650 minus N end fraction right parenthesis equals 1.2 end cell row blank cell not stretchy rightwards double arrow fraction numerator 350 over denominator 650 minus N end fraction equals e to the power of 6 over 5 end exponent not stretchy rightwards double arrow fraction numerator 650 minus N over denominator 350 end fraction equals e to the power of negative 6 over 5 end exponent end cell row blank cell N equals 650 minus 350 e to the power of negative 6 over 5 end exponent almost equal to 545 end cell end table en

إذن، بعد ثلاث سنوات يكون عدد الذئاب في تلك الغابة 545 ذئباً نقريباً.

كرة: تنكم ش كرة، ويتغيّر نصف قطرها بمعدل يمكن نمذجتـه بالمعادلة التفاضلية: begin mathsize 20px style fraction numerator bold d bold r over denominator bold d bold t end fraction bold equals bold minus bold 0 bold. bold 0075 bold italic r to the power of bold 2 end style، حيث begin mathsize 20px style bold italic r end style طول نصف قطر الكرة بالسنتيمتر، وbegin mathsize 20px style bold italic t end style الزمن بالثواني بعد بدء انكماش الكرة:

(27) أحل المعادلة التفاضلية لإيجاد طول نصف قطر الكرة بعد begin mathsize 20px style t end style ثانية، علماً بأن طول نصف الكرة الابتدائي هو begin mathsize 20px style 20 space c m end style.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d r over denominator d t end fraction equals negative 0.0075 r squared end cell row blank cell integral negative fraction numerator d r over denominator r squared end fraction equals integral 0.0075 d t end cell row blank cell 1 over r equals 0.0075 t plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 1 over 20 equals 0 plus C not stretchy rightwards double arrow C equals 1 over 20 space open parentheses t equals 0 comma straight r equals 20 space نعوض close parentheses end cell row blank cell 1 over r equals 0.0075 t plus 1 over 20 not stretchy rightwards double arrow r equals fraction numerator 20 over denominator 1 plus 0.15 t end fraction space open parentheses الخاص space الحل close parentheses end ce

(28) بعد كم ثانية يصبح طول نصف قطر الكرة begin mathsize 20px style 10 space c m end style؟

نضع r=10 في المعادلة الناتجة: 

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 10 equals fraction numerator 20 over denominator 1 plus 0.15 t end fraction not stretchy rightwards double arrow 0.1 equals fraction numerator 1 plus 0.15 t over denominator 20 end fraction end cell cell not stretchy rightwards double arrow 2 equals 1 plus 0.15 t end cell row blank cell not stretchy rightwards double arrow t equals fraction numerator 1 over denominator 0.15 end fraction almost equal to 6.67 straight s end cell end table end style

إذن، يكون طول نصف قطر الكرة 10cm بعد 6.67 ثانية تقريباً بعد بدء انكماشها.

حشرات: يتغير عدد الحشرات في مجتمع للحشرات بمعدل يمكن نمذجته بالمعادلة التفاضلية: begin mathsize 20px style fraction numerator bold d bold n over denominator bold d bold t end fraction bold equals bold 0 bold. bold 2 bold italic n bold left parenthesis bold 0 bold. bold 2 bold minus bold italic c bold italic o bold italic s bold invisible function application bold italic t bold right parenthesis end style، حيث begin mathsize 20px style bold italic n end style عدد الحشرات، وbegin mathsize 20px style bold italic t end style الزمن بالأسابيع بعد بدء ملاحظة الحشرات:

(29) أحل المعادلة التفاضلية لإيجاد عدد الحشرات في هذا المجتمع بعد begin mathsize 20px style t end style أسبوعاً، علماً بأن عددها الابتدائي هو 400 حشرة.

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell integral fraction numerator d n over denominator n end fraction equals integral 0.2 left parenthesis 0.2 minus c o s invisible function application t right parenthesis d t end cell row blank cell l n invisible function application n equals 0.2 left parenthesis 0.2 t minus s i n invisible function application t right parenthesis plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell l n invisible function application 400 equals 0 plus C not stretchy rightwards double arrow C equals l n invisible function application 400 space open parentheses t equals 0 comma n equals 400 space نعوض close parentheses end cell row blank cell l n invisible function application n equals 0.2 left parenthesis 0.2 t minus s i n invisible function application t right parenthesis plus l n invisible function application 400 end cell row blank cell not stretchy rightwards double arrow l n invisible function application n over 400 equals 0.2 left parenthesis 0.2 t minus s i n invisible function application t right parenthesis not stretchy rightwards double arrow n equals 400 e to the power of 0.2 left parenthesis 0.2 t minus s i n invisible function application t right parenthesis end exponent space open parentheses الخاص space الحل close parentheses end ce

(30) أجد عدد الحشرات في هذا المجتمع بعد 3 أسابيع.

نعوض t=3 في المعادلة الأخيرة: 

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell n equals 400 e to the power of 0.2 left parenthesis 0.2 t minus s i n invisible function application t right parenthesis end exponent end cell row blank cell equals 400 e to the power of 0.2 left parenthesis 0.6 minus s i n invisible function application 3 right parenthesis end exponent end cell row blank cell almost equal to 400 e to the power of 0.12 minus 0.028 end exponent almost equal to 400 e to the power of 0.092 end exponent almost equal to 439 end cell end table end style

إذن، بعد 3 أسابيع يكون عدد الحشرات 439 حشرة تقريباً.

(31) تمثل المعادلة التفاضلية: begin mathsize 20px style fraction numerator d y over denominator d x end fraction equals y cos invisible function application x end style ميل المماس لمنحنى علاقة ما، أجد قاعدة هذه العلاقة إذا علمت أن منحناها يمر بالنقطة (0,1).

begin mathsize 20px style table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator d y over denominator d x end fraction equals y c o s invisible function application x end cell row blank cell integral fraction numerator d y over denominator y end fraction equals integral c o s invisible function application x d x end cell row blank cell not stretchy rightwards double arrow l n invisible function application vertical line y vertical line equals s i n invisible function application x plus C space open parentheses العام space الحل close parentheses end cell 
d table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 0 equals 0 plus C not stretchy rightwards double arrow C equals 0 space open parentheses y equals 1 comma x equals 0 space نعوض close parentheses end cell row blank cell not stretchy rightwards double arrow l n invisible function application vertical line y vertical line equals s i n invisible function application x end cell row blank cell not stretchy rightwards double arrow y equals e to the power of s i n invisible function application x end exponent end cell end ta

ملاحظة: منحنى الاقتران begin mathsize 20px style y equals negative e to the power of s i n invisible function application x end exponent end style لايمر بالنقطة (0,1).

(32) تمثل المعادلة التفاضلية: begin mathsize 20px style x left parenthesis x plus 1 right parenthesis fraction numerator d y over denominator d x end fraction equals y end style ميل المماس لمنحنى علاقة ما، أجد قاعدة هذه العلاقة إذا علمت أن منحناها يمر بالنقطة (1,3). 

begin mathsize 20px style integral fraction numerator d y over denominator y end fraction equals integral fraction numerator d x over denominator x left parenthesis x plus 1 right parenthesis end fraction space open parentheses الجزئية space الكسور space نستخد�
close parentheses\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 1 over denominator x left parenthesis x plus 1 right parenthesis end fraction equals A over x plus fraction numerator B over denominator x plus 1 end fraction end cell row blank cell A left parenthesis x plus 1 right parenthesis plus B left parenthesis x right parenthesis equals 1 end cell row blank cell x equals 0 not stretchy rightwards double arrow A equals 1 end cell row blank cell x equals negative 1 not stretchy rightwards double arrow B equals negative 1 end cell row blank cell not stretchy rightwards double arrow fraction numerator 1 over denominator x left parenthesis x plus 1 right parenthesis end fraction equals 1 over x plus fraction numerator negative 1 over denominator x plus 1 end fraction end cell row blank cell not stretchy rightwards double arrow integral fraction numerator d y over denominator y end fraction equals integral fraction numerator d x over denominator x left parenthesis x plus 1 right parenthesis end fraction not stretchy rightwards double arrow ln invisible function application vertical line y vertical line equals integral left parenthesis 1 over x plus fraction numerator negative 1 over denominator x plus 1 end fraction right parenthesis d x end cell row blank cell not stretchy rightwards double arrow ln invisible function application vertical line y vertical line equals ln invisible function application vertical line x vertical line minus ln invisible function application vertical line x plus 1 vertical line plus C space open parentheses العام space الحل close pa
ntheses end cell end table\ntable attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell ln invisible function application 3 equals 0 minus ln invisible function application 2 plus C not stretchy rightwards double arrow C equals ln invisible function application 3 plus ln invisible function application 2 equals ln invisible function application 6 space open parentheses y equals 3 comma x equals 1 space نعوض close parentheses end cell row blank cell not stretchy rightwards double arrow ln invisible function application vertical line y vertical line equals ln invisible function application vertical line x vertical line minus ln invisible function application vertical line x plus 1 vertical line plus ln invisible function application 6 end cell row blank cell not stretchy rightwards double arrow ln invisible function application vertical line y vertical line equals ln invisible function application vertical line fraction numerator 6 x over denominator x plus 1 end fraction vertical line end cell row blank cell not stretchy rightwards double arrow vertical line y vertical line equals vertical line fraction numerator 6 x over denominator x plus 1 end fraction vertical line not stretchy rightwards double arrow y equals fraction numerator 6 x over denominator x plus 1 end fract

ملاحظة: منحنى الاقتران begin mathsize 20px style y equals negative fraction numerator 6 x over denominator x plus 1 end fraction end style لا يمر بالنقطة (1,3)

إعداد : شبكة منهاجي التعليمية

10 / 07 / 2023

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