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- Homework Statement
- psb

- Relevant Equations
- psb

I tried to follow the method outlined in lectures, and ended up with an incorrect solution. My understanding of PDEs is a bit shaky so I thank anyone for constructive feedback or information.

The solution to the Poisson equation

\begin{equation}

-\Delta u(x)=\frac{q}{\pi a^3}e^{-\frac{2||x||}{a}}

\end{equation}

is given by

\begin{equation}

u(x)=\int_{R^3}\Phi(x-y)f(y)dy

\end{equation}

in ##R^3## and spherically symmetric ##f(y)## this is

\begin{equation}

u(x)=\frac{1}{2||x||}\int_0^\infty r\tilde{f}(r)\Big(||x||+r-\Big|||x||-r\Big|\Big)dr

\end{equation}

\begin{equation}

\frac{1}{2||x||}\int_0^\infty r\frac{q}{\pi a^3}e^{-\frac{2||x||}{a}}\Big(||x||+r-\Big|||x||-r\Big|\Big)dr

\end{equation}

\begin{equation}

\frac{1}{||x||}\frac{q}{\pi a^3}\int_0^\infty e^{-\frac{2r}{a}}r^2dr

\end{equation}

after integration by parts

\begin{equation}

u(x)=\frac{1}{||x||}\frac{q}{4\pi}

\end{equation}