Let y=y(x) be the solution of the differential equation dy dx + 2xy = 2x^3 with the initial condition y(0) = 0 . Then the value of _ x 0 y(x) x^4 is equal to

Step-by-step solution

Given differential equation: dy dx + 2xy = 2x^3 This is a linear differential equation with P(x) = 2x and Q(x) = 2x^3 . The integrating factor (IF) is: IF = e^ 2x dx = e^ x^2 The general solution is: y e^ x^2 = 2x^3 e^ x^2 dx + C To evaluate the integral, let t = x^2 , then dt = 2x dx . x^2 e^ x^2 (2x dx) = t e^t dt Using integration by parts: t e^t dt = t e^t - e^t = x^2 e^ x^2 - e^ x^2 Thus, the solution is: y e^ x^2 = x^2 e^ x^2 - e^ x^2 + C Using the initial condition y(0) = 0 : 0 = 0 - 1 + C C = 1 So, y e^ x^2