Use the method of characteristics to solve the PDE system
We begin by parameterizing as functions of such that
This gives the system of ODE
Unfortunately, we don’t know what is; but we do have the parameterization parameter , so we can solve the system backwards in time.
which is equivalent to the second order ODE
which has the solution
Now, we know that , so we want to find the time such that , or , then
Let’s quickly verify that this is indeed the solution
Since this is fully linear, the solution is unique and the characteristics intersect.