Originally Posted by

**Silverflow** Hi there,

I have a question that states:

Obtain the general solution of $\displaystyle L[y] =ty'' + (2-t)y' -y = e^t$ by factorizing L into two first order linear operators.

My thoughts on what to do, is it divide through by t, then make $\displaystyle z_{1}$= y, $\displaystyle z_{2}$ = y' and $\displaystyle z_{3}$ = y''.

That makes $\displaystyle z_{3}= -\frac{2-t}{t}z_{2} -\frac{1}{t} z_{1} +\frac{e^t}{t}$. Then solve for this first order DE. Once finding a solution to that DE, replace y'' with the answer, than solve the first order linear equation then.

I was wondering whether my thought process was right. Is my method correct?

Thanks for your time.