Solution
= Solution
Taking the Laplace transform in $t$ gives
$$
U_{xx}-sU=-f(x).
$$
The bounded resolvent solution supplied in the hint can be written
$$
U(x,s)=\int_{-\infty}^{\infty}\frac{e^{-\sqrt s|x-\xi|}}{2\sqrt s}f(\xi)\,d\xi.
$$
Part (a) inverts the kernel, yielding the <heat kernel>
$$
\boxed{K(|x-\xi|,t)=\frac1{\sqrt{4\pi t}}\exp\left(-\frac{(x-\xi)^2}{4t}\right).}
$$