Let y = y(x) be the solution curve of the differential equation x dy dx + y y = x y e^x for x > 0 and y > 0 . If y(1) = e , then y(2) is equal to

Step-by-step solution

Given differential equation: x dy dx + y y = x y e^x Dividing the entire equation by xy , we get: 1 y dy dx + 1 x y = e^x Let y = v . Differentiating with respect to x , we get: 1 y dy dx = dv dx Substituting this into the equation, we obtain a linear differential equation in v : dv dx + 1 x v = e^x Here, P(x) = 1 x and Q(x) = e^x . Integrating Factor (I.F.) = e^ 1 x dx = e^ x = x The general solution is given by: v x = (e^x x) dx + C Using integration by parts: v x = x e^x - 1 e^x dx + C x y = x e^x - e^x + C Give