I need helping solving this Inital Value problem!

Question

I'm trying to find the solution to this IVP problem:

$$\frac{dy}{dt} = \frac{2}{t}y + t^2e^t$$

$$ 1\leq t \leq2$$ $$y(1) = 2$$

I'm not entire sure how to go about solving this problem.

Answer 1

First of all you have both $t$ and $x$ in your equation which needs attention.

Assuming that your equation is $$\frac {dy}{dt}=\frac {2}{t}y+t^2e^t$$ it is a linear equation which is solved by integrating factors method

The integrating factor is $t^{-2}$ and your equation turns into $$t^{-2} \frac {dy}{dt}-2t^{-3}y=e^t$$

Which is integrable $$t^{-2}y=e^t+c$$