This is an exercise from the book Partial differential equations by Robert C. McOwen:
Consider the wave PDE $$ u_{tt} − u_{xx} + \lambda u = 0,\ x \in \mathbb R, t > 0 $$
Find two uniform wave solutions with $\lambda > 0$, satisfying the I.C. $u(x, 0) = 3 \cos 2x$
From my knowledge, for the uniform solution, one can write $$u(x,t)=U(kx-\omega t)=Ae^{i(kx-\omega t)}$$
Then if we substitute the above equation into the given one and plug in the initial condition, we'll get $$u(x,t)=3\cos(2x-\omega t)+3i\sin(2x-\omega t)$$ which doesn't agree with the given I.C. I don't know where I went wrong. Thanks