Question

find the general solution of the differential equation x dy/dx + 4y =x^3

Answer 1

Let us find the general solution of the differential equation $x \frac{dy}{dx} + 4y =x^3$. For this let us multiply both parts by $x^3:$ $x^4y' + 4x^3y =x^6$. The latter equation is equivalent to $(x^4y)'=x^6$. It follows that $x^4y=\frac{x^7}{7}+C,$ and hence the general solution is $y=\frac{x^3}{7}+\frac{C}{x^4}.$