Find all solutions of the form $u(x, t) = \phi(x−ct)$ (so called travelling waves) for the partial differential equation $$u_t = u_{xx} + u$$
I just had my first two lectures on partial differential equations and the explanations from our lecturer are not really working for me. Can someone show me how to solve a partial differential equation like the one in my question, or even better, refer me to a good source for beginners? Does the equation I mentioned above have a special name or a special "ansatz" to find solutions?
Substitute the suggested form into the equation. You should get
$$-c\phi' = \phi'' + \phi$$
which is an ODE. I hope you can solve it. Just keep in mind that a solution is a function of $x - ct$.