Differential equations with distributions

1. Jan 25, 2012

tjackson3

1. The problem statement, all variables and given/known data

Solve $x^2\frac{du}{dx} = 0$ in the sense of distributions.

2. Relevant equations

<u',f> = -<u,f'> for any test function f.

3. The attempt at a solution

My thinking is that since we want to see the action of the left hand side on a general test function f, we try

<x^2u',f> = -<u,(x^2f)'> = 0

so clearly we can drop the negative

<u,(x^2f)> = 0

But I'm stuck as to where to go from here. On one hand, it would seem like the delta function would satisfy this, since $\int_{-\infty}^{\infty} \delta(x)x^2f(x)\ dx = 0$. However, besides the fact that this seems too easy, there's an example in the book I'm using (Keener) which shows that the solution for the differential equation $x\frac{du}{dx} = 0$ is $c_1 + c_2H(x)$. Since the solution for $\frac{du}{dx} = 0$ is just the constant distribution, this would seem to imply to met that the solution to my differential equation involves the constant distribution, the Heaviside distribution, and some other distribution. The book is extremely unclear on how to solve these problems. Any tips?

Thanks!