Solve the Wave Equation
I've been trying to solve the above wave equation where $u = u(x, t)$ and $c ∈ \mathbb{R}$ is a constant, subject to $$ u(x, 0) = 0,\;\; 0 < x < 1, \\ u_t(x, 0) = U_0x,\;\; 0 < x < 1, \\ u(0, t) = 0, \;\; t > 0, \\ u(1, t) = 0, \;\; t > 0. $$
I know that a Sturm-Liouville Boundary Value Problem for a function $y(x)$ on some interval $a ≤ x ≤ b$ is a two-point boundary value problem which satisfies the ODE but am unsure of how to use the information given to me to solve this wave equation as a Sturm-Liouville BVP.
After separation of variables, and enforcing homogeneous conditions, $$ u(x,t) = \sum_{n=1}^{\infty}A_n \sin(n\pi c t)\sin(n\pi x). $$ The condition $u_t(x,0)=U_0 x$ determines the constants $A_n$: $$ U_0 x = \sum_{n=1}^{\infty}A_nn\pi c\sin(n\pi x) $$ The coefficients are determined by orthogonality of the functions $\sin(n\pi x)$: $$ U_0\int_{0}^{\pi}x\sin(n\pi x)dx = A_n n\pi c\int_{0}^{\pi}\sin^2(n\pi x)dx \\ A_n = \frac{U_0}{n\pi c}\cdot \frac{\int_{0}^{\pi}x\sin(n\pi x)dx}{\int_0^{\pi}\sin^2(n\pi x)dx}. $$