Let y = y(x) be the solution of the differential equation dy dx + x y = x^3 satisfying the initial condition y(0) = 1 . Then the value of _ x y(x) x^2 is

Step-by-step solution

The given differential equation is dy dx + x y = x^3 . This is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = x and Q(x) = x^3 . The integrating factor (I.F.) is e^ x dx = e^ x^2/2 . Multiplying both sides by the I.F. and integrating, we get: y e^ x^2/2 = x^3 e^ x^2/2 dx To evaluate the integral, let x^2 = t 2x dx = dt . Then x^3 e^ x^2/2 dx = x^2 e^ x^2/2 x dx = t e^ t/2 dt 2 . Using integration by parts: 1 2 [ t e^ t/2 1/2 - 1 e^ t/2 1/2 dt ] = t e^ t/2 - 2 e^ t/2 = (x^2 - 2)e^ x^2