= Solution
{c}
Enforce the fixed-end <Dirichlet boundary condition> by taking the <odd extension> of the initial <velocity> to the whole line:
$$
v_o(s)=\kappa s e^{-|s|/L}.
$$
This agrees with the specified <initial condition> for $s\geq0$. To derive the relevant form of <D'Alembert's formula>, the zero initial displacement in <b> gives $G=-F$, up to a cancelling constant. The initial <velocity> then gives $-2cF'=v_o$. Hence
$$
y(x,t)=\frac1{2c}\int_{x-ct}^{x+ct}v_o(s)\,ds.
$$
An <antiderivative>, valid across $s=0$, is
$$
\int s e^{-|s|/L}\,ds=-L(|s|+L)e^{-|s|/L}.
$$
Evaluation therefore gives the solution throughout $x>0,t>0$:
$$
\boxed{y(x,t)=\frac{\kappa L}{2c}
\left[(|x-ct|+L)e^{-|x-ct|/L}
-(x+ct+L)e^{-(x+ct)/L}\right].}
$$
For $ct\leq x$, replace $|x-ct|$ by $x-ct$; for $ct\geq x$, replace it by $ct-x$. Thus the same expression includes the reflected <wave> after it reaches the fixed end.
At $x=0$ the two terms agree, so $y(0,t)=0$. At $t=0$ they again agree, so $y(x,0)=0$. Differentiating the <integral> with respect to $t$ gives
$$
y_t(x,t)=\frac12\bigl[v_o(x+ct)+v_o(x-ct)\bigr],
$$
which recovers $y_t(x,0)=\kappa xe^{-x/L}$. Since $v_o$ is a <continuously differentiable function>, the formula is twice continuously differentiable even across $x=ct$, and satisfies the <wave equation> there as well.
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