Solve a differential equation on distribution

Question

I don't know how to solve $U'=Uz$ where $U$ is a distribution and $z$ is complex. Thanks for your help!

Answer 1

First, let us take a look at the solution of this equation as if it were a regular ODE: obviously, the solution would be $t\to \exp(tz)$ up to a constant factor.

Since this function is $C^\infty$ and can never be zero, we can safely divide our distributional equation by this function: $$e^{-tz}U'-ze^{-tz}U=0.$$ You might notice that in the left hand side we have a derivative: $$\frac d{dz} \left(e^{-tz}U\right)=0.$$ It implies that $e^{-tz}U = A$ for a constant $A$; for the same reasons as above, we can multiply this identity by $e^{tz}$ without losing or generating distributional solutions: $$U=Ae^{tz},\quad A\in\Bbb C.$$

Hence, the conclusion - in some good cases distributional differential equations give nothing beyond solutions in the usual sense.