Let y=y(x) be the solution of the differential equation (2 x _e x ) d y d x +2 y= 3 x _e x, x>0 and y (e⁻¹ )=0 . Then, y(e) is equal to

Step-by-step solution

aligned & d y d x + y x x = 3 2 x^2 & I.F. =e^ 1 x x d x =e^ ( (x)) = x & y x= 3 x 2 x^2 d x & = 3 x 2 x⁻² d x- ( 3 2 x x⁻² d x ) d x & = 3 x 2 (- 1 x )- 3 2 x (- 1 x ) d x & y. x= -3 n x 2 x - 3 2 x +C aligned aligned & y ( e ⁻¹ )=0 & 0(-1)= 3 e 2 - 3 e 2 + C C =0 & y = -3 nx 2 x - 3 2 x & y ( e )= -3 2 e - 3 2 e = -3 e aligned