Let y=y(x) be the solution curve of the differential equation y d y ~d x +2 x y=x^3 y, y(1)=0 . Then y( 3 ) is equal to :

Step-by-step solution

aligned & ^2 y d y d x +2 x y secy =x^3 y secy & ^2 y d y d x +2 x y=x^3 & tany =t ^2 y d y d x = d t d x & d t d x +2 x t=x^3, If =e^ 2 x d x =e^ x^2 & t x^ x^2 = x^3 e^ x^2 d x+c & x^2=Z t e^Z= 1 2 e^Z Z d Z= 1 2 [e^Z Z-e^Z ]+c & 2 y= (x^2-1 )+2 e^ -x^2 & y(1)=0 c=0 y( 3 )= 4 aligned