TITLE: Solve the following Cauchy Euler equation:
QUESTION [0 upvotes]: How to solve the following Cauchy-Euler equation:
$$x^2y''-xy'+y = \ln(x)$$
In class we have solved only homogenous equations, thus I'm not particulary sure how to do this. I tried setting $y=x^\lambda$, but did not seem to work. The logrithm on the right messes up my stuff. I also differentiated with respect to $x$ to remove it and got to the equation:
$$x^3y''' + x^2y'' = 1$$
Which again I can't solve.

REPLY [1 votes]: Let $x=e^t$ and then transform the equation to
$$\ddot y-2\dot y+y=t$$ and then solve it
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