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My attempt at a solution for $ u(x,t) $ is below, I don't think my $ G(t) $ and $ \phi(x) $ can satisfy the initial condition $ \dfrac{\mathrm du}{\mathrm dt}(x,0)=4 + 3\cos(2 \pi x) $ when DSolve is giving me Cosh and Sinh but it's very likely that I'm using DSolve incorrectly even though after wrestling with it for hours.

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 (* 11.3.0 for Microsoft Windows (64-bit) (March 7, 2018) *)

 pde = D[u[x, t], t, t] == D[u[x, t], x, x] - 4*u[x, t];
 icb = {Derivative[1, 0][u][0, t] == 0, Derivative[1, 0][u][1, t] == 0,
 u[x, 0] == 0, Derivative[0, 1][u][x, 0] == 5 + 3*Cos[2 Pi x]};

 sol = DSolve[{pde, icb}, u, {x, t}]

 (* {{u -> Function[{x, t}, 5/2 Sin[2 t] + (3 Cos[2 \[Pi] x] Sin[2 Sqrt[1 + \[Pi]^2] t])/
 (2 Sqrt[1 + \[Pi]^2])]}}*)

$$u(x,t)=\frac{5}{2} \sin (2 t)+\frac{3 \cos (2 \pi x) \sin \left(2 \sqrt{1+\pi ^2} t\right)}{2 \sqrt{1+\pi ^2}}$$

Check pde equation:

 pde /. sol // FullSimplify
 (* {True} *)(*OK*)

Check initial and boundary conditions: icb:

 icb /. sol
 (* {{True, True, True, True}} *)(*OK*)
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