If d y d x + e x x 2 − 2 y = x 2 − 2 x x 2 − 2 e 2 x and y 0 = 0 , then the value of y 2 is

Step-by-step solution

Given d y d x + e x x 2 − 2 y = x 2 − 2 x x 2 − 2 e 2 x It is linear differential equation so, I F = e ∫ x 2 - 2 e x d x = e x 2 e x - 2 ∫ x e x d x - 2 e x = e x 2 e x - 2 x e x - e x - 2 e x I F = e x 2 - 2 x e x Now solution is given by y × e x 2 - 2 x e x = ∫ e x 2 - 2 x e x × x 2 - 2 x x 2 - 2 e 2 x = ∫ e x 2 - 2 x e x × x 2 - 2 x e x x 2 - 2 e x Now let x 2 - 2 x e x = t We get e x x 2 - 2 x + e x 2 x - 2 d x = d t e x x 2 - 2 x + 2 x - 2 d x = d t e x