Could you give me some hints about the question in the picture?
The solution Method 1: I know we can solve by using Separation of variables Method.
The solution method 2: I will take derivative $u(x,t)$ with respect to $t$ and second derivative w.r.t. $x$, then will substitute them $u_t=c^2u_{xx}$.
The solution method 2 is right?
Thank you.
If this is for homework it depends what your instructor wants. Given a solution it seems to be fine to show that it satisfies the equation by plugging it in, however, it is not always straight forward to take derivatives of series (properly anyways), although you should be fine in this case. You can take the derivatives and plug it into the equation and also make sure it satisfies the boundary conditions to show that this is indeed a solution.
However, if I were you, to play it safe, I would do separation of variables and derive the answer given. It is not too difficult to do if you know the method!