Hi ! So I'm having trouble with a problem asking to find the integrating factor of this equation :

$\displaystyle t^{2}(\frac{dx}{dt})+t^{5}x = 4t $

So my approach was to set the right-hand side to 0 to get a homogeneous DE and then use separation of variables.

This got me :

$\displaystyle \frac{dx}{dt} = -t^{5}x/t^{2} = -t^{3}x$

$\displaystyle \frac{dx}{x} = -t^{3}dt$

$\displaystyle x = ce^{\frac{-t^{4}}{4}}$

So the general solution to the inhomogeneous one with u as integrating factor is : x= $\displaystyle ue^{\frac{-t^{4}}{4}}$

But plugging this into the original equation I arrive at :

$\displaystyle u'=\frac{4e^{\frac{-t^{4}}{4}}}{t}$

Which I don't know how to integrate and doesn't look like it can be any of the answers I was offered (multiple-choice test).

Where did I go wrong ?

Thank you for your help !