I was working on the following generalization of the wave equation;
$u_{tt}-c^2u_{xx}+m^2u=0$
My professor had suggested a technique similar to d'Alembert's principle would work but I see no useful way of factorising $(\frac{\partial}{\partial t^2}-c^2\frac{\partial}{\partial x^2}+m^2)$. I also tried using the function $e^{mt}u$ but to no avail. Please suggest some way of doing this.
Thanks in advance.
Let's write $u(x,t)$ as $$ u(x,t)=\int_{-\infty}^{\infty}U(k,t)e^{ikx}\,dk. \tag{1} $$ Plugging $(1)$ into the PDE we conclude that $U$ must satisfy the ODE $$ U_{tt}+(c^2k^2+m^2)U=0, \tag{2} $$ which has as general solution $$ U(k,t)=A(k)e^{i\omega_kt}+B(k)e^{-i\omega_kt}\qquad \left(\omega_k=\sqrt{c^2k^2+m^2}\right), \tag{3} $$ hence $$ u(x,t)=\int_{-\infty}^{\infty}\left[A(k)e^{i(kx+\omega_kt)}+B(k)e^{i(kx-\omega_kt)}\right]dk. \tag{4} $$