Differential Equations initial value problem

[hocuspocus102]

Homework Statement

Let f(t) be the solution to the initial value problem 2t(dy/dt)+y=t^4
with f(0) = 0
find f(t).

Homework Equations

The Attempt at a Solution

I tried to do this by separating variables but that hasn't gotten me very far. I don't know if I can do it by doing that thing where you find an integrating factor and like raising e^(some integral)? It didn't really make any sense to me when I tried it, so could someone explain it please? Thanks!

 

[arkajad]

Or a cleaver substitution? Something of the form t^\alpha y with some handy value of \alpha? Does not the left hand side of your equation look like a derivative of a something?

 

[HallsofIvy]

Because it is a linear equation, there is a formula for the "integrating factor". Write it as dy/dx+ y/2t= (1/2)t^3.

Now look for u(t) such that d(uy)/dt= u (dy/dt)+ (du/dt)y= u dy/dt+ (u/2t)y