Let y = y(x) be the solution of the differential equation (1+x^2) dy dx + 2xy = 4x 1+x^2 with the initial condition y(0) = 0 . The maximum value of y(x) for x R is

Step-by-step solution

First, rewrite the given differential equation in standard linear form by dividing by (1+x^2) : dy dx + 2x 1+x^2 y = 4x (1+x^2)^2 The integrating factor (I.F.) is: I.F. = e^ 2x 1+x^2 dx = e^ (1+x^2) = 1+x^2 Multiplying the standard equation by the I.F., we get: d dx (y(1+x^2) ) = 4x 1+x^2 Integrating both sides with respect to x : y(1+x^2) = 4x 1+x^2 , dx y(1+x^2) = 2 (1+x^2) + C Using the initial condition y(0) = 0 : 0(1+0) = 2 (1) + C C = 0 Thus, the explicit solution is: y(x) = 2 (1+x^2) 1+x^2 To find the maximu