I've been mulling over a problem from Friedlander's Introduction to Distribution Theory for a few days now: in Chapter 3 (on distributions with compact support), it asks to solve the differential equation $x\partial u-\lambda u=0$ for arbitrary $\lambda\in\mathbb{C}$.
I've seen it stated that a distribution $u\in\mathbb{R}^n$ is homogeneous of degree $\lambda$ iff it satisfies Euler's equation, i.e. $\sum x_i\partial_i u=\lambda u$, but these just refer to Gel'fand and Shilov's book which I don't have access to. Moreover, this approach doesn't seem the author's intended one, given the chapter this problem is in.
My method was basically to try and use "integrating factors", but of course we have only defined multiplication by smooth functions, which $x_+^\lambda$ is generally not. I've basically seen it without proof via online searching that $u=Ax_+^\lambda+Bx_-^\lambda$ is the general answer, but I can't seem to crack it, or find anything in this problem with compact support. Any help would be much appreciated.