Problem: Solve the following differential equation by using the method of characteristics.
\begin{align*}
\frac{\partial u}{\partial t} + 5 \frac{\partial u}{\partial x} = 0, & \quad -\infty \lt x \lt \infty , \ t > 0, \\
u(x,0) = e^{-x^2}.
\end{align*}
Solution: The first characteristic equation is
\[
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = 5 \implies x(t) = 5t + c \implies c = x(t) - 5t.
\]
Thus,
\begin{align*}
u(x(t), t) = f(c) = e^{-c^2} = e^{-\left[ x(t) - 5t \right] ^2}.
\end{align*}
Thus, the solution of the given PDE is,
\[
u(x,t) = e^{-(x-5t)^2}.
\]