Exercise 104

Use the Laplace transform in $t$ to solve $$\begin{aligned} & y_{tt} = y_{xx}, \qquad -\infty < x < \infty, \enspace t > 0,\\ & y(x,0) = \sin(x), \quad y_t(x,0) = 0 . \end{aligned}$$ Assume that $y$ is bounded. Hint: Note that for any fixed $s > 0$, $e^{sx}$ blows up as $x \to +\infty$ and $e^{-sx}$ blows up as $x \to -\infty$.