The general solution of the differential equation dy dx + y x = x^2 + 5 is ....

Step-by-step solution

The given differential equation is dy dx + 1 x y = x^2 + 5 . This is a linear differential equation of the form dy dx + Py = Q , where P = 1 x and Q = x^2 + 5 . The integrating factor is e^ P dx = e^ 1 x dx = e^ x = x . Multiplying the equation by the integrating factor, the solution is given by: y x = x(x^2 + 5) dx + C xy = (x^3 + 5x) dx + C xy = x^4 4 + 5x^2 2 + C Rearranging the terms, we get: x^4 4 + 5x^2 2 - xy = -C Taking -C = c , the general solution is: x^4 4 + 5x^2 2 - xy = c Answer: x^4 4 + 5x^2 2 - xy =