How to solve this Boundary Value Problem? (Using Fourier)

Question

utt=uxx−u$u_{tt}=u_{xx}-u$
Boundary condition: u(x,0)=f(x),ut(x,0)=0,$u(x,0)=f(x),u_t(x,0)=0,$ f(x)$f(x)$ is a Schwartz function.

I tried to make u(x,t)=X(x)T(t)$u(x,t)=X(x)T(t)$, then I get XX′′=TT′′+T$\frac{X}{X^{″}}=\frac{T}{T^{″}+T}$.
Let XX′′=λ$\frac{X}{X^{″}}=\lambda$, then X′′=λX$X^{″}=\lambda X$ and T′′+(1−λ)T=0$T^{″}+(1-\lambda )T=0$.
Then I stuck on that. What should I do next?