$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Manufactured solution: example
$$ \uex(x,t) = x(L-x)\sin t$$
PDE \( u_{tt}=c^2u_{xx}+f \):
$$ -x(L-x)\sin t = -2\sin t + f\quad\Rightarrow f = (2 - x(L-x))\sin t$$
Implied initial conditions:
$$
\begin{align*}
u(x,0) &= I(x) = 0\\
u_t(x,0) &= V(x) = - x(L-x)
\end{align*}
$$
Boundary conditions:
$$ u(x,0) = u(x,L) = 0 $$
