$u_{tt} = u_{xx} - u$
Boundary condition: $u(x,0)=f(x), u_t(x,0)=0,$ $f(x)$ is a Schwartz function.
I tried to make $u(x,t)=X(x)T(t)$, then I get $\frac{X}{X''}=\frac{T}{T'' + T}$.
Let $\frac{X}{X''}=\lambda$, then $X''=\lambda X $ and $T'' + (1-\lambda )T=0$.
Then I stuck on that. What should I do next?