If y = y x is the solution of the differential equation x d y d x + 2 y = x e x , y 1 = 0 then the local maximum value of the function z x = x 2 y ( x ) - e x , x ∈ R is

Step-by-step solution

Given, d y d x + 2 y x = e x I . F . = e ∫ 2 x d x = e 2 ln x = x 2 ∴ General solution: y x 2 = ∫ e x x 2 d x We know that ∫ e x f x d x = = e x f x - f ' ( x ) + f '' x - f ''' x + ⋯ + - 1 n f n x + C So, y x 2 = e x x 2 - 2 x + 2 + C             . . . i Given y 1 = 0 ⇒ 0 1 = e 1 1 - 2 1 + 2 + C ⇒ C = - e ∴ y = e x x 2 x 2 - 2 x + 2 - e (from eq i ) Hence, z ( x ) = x 2 e x x 2 x 2 - 2 x + 2 - e - e x = e x x 2 -