Solution of the PDE $yu_x+xu_y=0$ subject to the initial condition $u(x,0) = \exp \left(-\frac{x^2}{2}\right)$ 
Consider the following first-order PDE $$yu_x+xu_y=0$$
subject to the initial condition $$u(x,0) = \exp \left(-\frac{x^2}{2}\right)$$ Show that the above problem has

*

*a unique solution in a neighbourhood of the point $(x_0,0)$ provided $x_0 \neq0,$


*infinitely many solution in a neighbourhood of origin.

Please someone give some hints. Thank you.

Using separation of variables, let
$$u (x,y) = X (x) Y (y)$$
The PDE can then be broken into $2$ uncoupled ODEs
$$\begin{array}{rl} \dfrac{X ' (x)}{X (x)} &= \,\,\,\,\gamma \, x\\\\ \dfrac{Y ' (y)}{Y (y)} &= -\gamma \, y \end{array}$$
Thus, the general solution is of the form
$$u (x,y) = u_0 \exp \left( \frac{\gamma}{2} \left( x^2 - y^2 \right) \right)$$
Using the given initial condition, the values of constants $u_0$ and $\gamma$ can be determined.
