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

Step-by-step solution

aligned & x (x^2+e^x ) d y+ (e^x(x-2) y-x^3 ) d x=0 & x (x^2+e^x ) d y d x +e^x(x-2) y=x^3 & d y d x + e^x(x-2) x (x^2+e^x ) y= x^2 x^2+e^x & I.F. =e^ e^x(x-2) x (x^2+e^x ) d x =e^ e^x ( 1 x^2 - 2 x^2 ) d x (1+ e^x x^2 ) d x aligned Let 1+ e^x x^2 =t x^2 e^x-e^x 2 x x^4 d x=d t I.F. e ^ (1+ e ^2 x ^2 ) =1+ e ^ x x ^2 Now y (1+ e^x x^2 )= x^2 x^2+e^x x^2+e^x x^2 d x+C y (1+ e^x x^2 )=x+C Passing through (1,0) aligned & C=-1 & y= x-1 1+ e^x x^2 & y(2)= 1 1+ e^2 4 = 4 4+e^2 aligned