Ex. 9: Find the value of \( n \) for which \( z=t^{n} e^{-r^{2} / 4 t} \) satisfies the partial differential equation \( \frac{1}{r^{2}}\left[\frac{\partial}{\partial r}\left(r^{2} \frac{\partial z}{\partial r}\right)\right]=\frac{\partial z}{\partial t} \).
Sol. : Given,
\[
z=t^{n} e^{-r^{2} / 4 t}
\]
