If y = y(x) is the solution of the differential equation dy dx + x e^x (x+1)^2 y = x e^ 2x (x+1)^3 satisfying y(0) = 0 , then the value of y(1) is

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + P(x)y = Q(x) . Here, P(x) = x e^x (x+1)^2 . The integrating factor (IF) is given by: IF = e^ P(x) dx = e^ x e^x (x+1)^2 dx We can rewrite the integral as: e^x ( x+1-1 (x+1)^2 ) dx = e^x ( 1 x+1 - 1 (x+1)^2 ) dx Using the standard result e^x (f(x) + f'(x)) dx = e^x f(x) , we get: P(x) dx = e^x x+1 Thus, IF = e^ e^x x+1 . The general solution is: y e^ e^x x+1 = x e^ 2x (x+1)^3 e^ e^x x+1 dx + C Let t = e^x x+1 . Then dt = e^x ( 1 x+