I am trying to solve $u_{tt} = u_{xx}, u(0,\ t) = 0, u(1,\ t) = 0, u(x,\ 0) = \frac{x - x^2}{4}, u_t(x,\ 0) = 0$. Using separation of variables leads to $\frac{\ddot T}{T}$ = $\frac{X''}{X}$ = $\lambda$. My solution to the Sturm-Louiville problems is $(X = Ae^{-\sqrt\lambda x} + Be^{\sqrt\lambda x}, T = Ce^{-\sqrt\lambda t} + De^{\sqrt\lambda t})$ if $\lambda \neq 0$ (I think I've ruled out the $\lambda = 0$ case). Thus I have $u = ACe^{-\sqrt\lambda x - \sqrt\lambda t} + ADe^{\sqrt\lambda t - \sqrt\lambda x} + BCe^{\sqrt\lambda x - \sqrt\lambda t} + BDe^{\sqrt\lambda x + \sqrt\lambda t}$.
It feels like I should be able to somehow lump constants to simplify the problem, but it is also more difficult than in the $1$D Heat Equation, where $T$ had only a single arbitrary constant. Here, due to the FOILing, there is a situation where if, i.e., $AC$, $AD$, and $BC$ are all positive, then $A$, $B$, $C$, and $D$ must all have the same sign, so that $BD$ cannot be negative, thus the exponential coefficients are not fully arbitrary. Can I simplify the constants at all here without modifying the constraints the coefficients impose on each other, as there seem to be too many constants left when I plug in the boundary conditions? My workflow when solving the Heat Equation was to switch from exponentials to trig via Euler's formula after plugging in the boundary conditions but before plugging in the initial conditions, after it was clear that $\lambda$ came out negative (it always did for the $1$D Heat Equation). I tried doing the conversion early here to see if constants would go away, by multiplying the exponents by $-i^2$ and applying Euler's formula, but the constants were merely scrambed rather than simplified.