I am having some issue with a conservation law, where there is a step source term. Anyway, I have the following PDE

Using method of characterstics and letting

we have

This implies that

.

From this point is where I get a little confused.

Our ODE is piecewise

.

So the solution to the ODE is

.

Which leads to the final solution

.

However, I know the solution is suppose to be

.