Math 6420 • PDE

\begin{enumerate}
\item[(a)] Show that a test function \(\psi(x)\) is of the form \(\psi=(x\phi)'\) where \(\phi\) is a
test function if and only if
\[
  \int_{-\infty}^{\infty}\psi(x)\,dx=0
  \quad\text{and}\quad
  \int_0^{\infty}\psi(x)\,dx=0.
\]

\item[(b)] Solve the following equation in the sense of distribution:
\[
  x^2\frac{d\phi}{dx}=0.
\]
\end{enumerate}