Solving PDEs with the Laplace transform

Solve

For a $c > 0$, solve

Find the corresponding ODE problem for $Y(x)$, after transforming the $t$ variable Do not solve the problem.

Use the Laplace transform in $t$ to solve 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 $x \to -\infty$.

Use the Laplace transform in $t$ to solve where $f(t)$ is some function. Assume that $y$ is bounded. Give the answer as a convolution.