Uniqueness of solution based on characteristic curves

Question

I have a pde $$\begin{cases} u_t − xu_x = 2u & x\in\mathbb{R}, t>0\\ u(x, 0) = \frac{1}{1+x^2} \end{cases}$$

I've solved it using method of characteristics ($u=\frac{1}{1+x^2e^{2t}}e^{2t})$ and plotted charactersitic curves.

Consider the upper half-space since $t>0$. How to argue using the drawing whether or not it is the unique solution? Thank you.

Answer 1

The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.