Given the equation, $$u_{xx}=\frac{1}{\kappa}u_t$$
How would I go about solving using Method of Characteristics? I know I end up with the PDE being transformed into something along the lines of $$u_{\phi\phi}+first~order~terms+other~terms=0$$ However, when trying to obtain $\phi$ and $\zeta$, I run into a few problems as I get to $$\frac{dt}{d x}=0$$ then have to solve $$\int dt=\int 0 \cdot dx$$
Any direction would be greatly appreciated.