solve the 1st order differential equation 2y+(x*dy/dx)=x^3

1st we note our three methods for solving 1st order differential equation: separating the x and y variables to both sides of the equation, and integrating both sides with respect to x. Recognising an exact differential equation gained from product rule, and using an integrating factor to gain exact differential equation and then solving this way. Observing our equation we note it is not separable, and nor is it exact given the derivative of x is not the 2 (the term which y is multiplied by), thus we must use an integrating factor and multiply the whole equation by this integrating factor.
To do this we must: Divide by x to get dy/dx on its own (giving (2/x)*y+(dy/dx)=(x²). We have the equation our integrating factor I(x)=e^(the integral with respect to x of r(x)) for differential equation y+r(x)dy/dx=p(x) for some functions r(x) and p(x). Therefore we integrate 2/x with respect to x we have I(x) (the integrating factor) =e^2lnx ,which by laws of logs equals x². Hence, multiplying through by our integrating factor we reach (dy/dx)(x²)+2xy=x⁴, which gives an exact differential equation equation (given when x²y is differentiated with respect to x by product rule we get the left side of our differential equation), so we may integrate both sides of equation with respect to x. Doing this gives us the equation x²y=x⁵/5 +C (C a constant of integration) and we may divide both sides of our equation by x² to give simplified answer y=x³/5+C/x²