Exercise 105

Use the Laplace transform in $t$ to solve $$\begin{aligned} & y_{t} = y_{xx}, \qquad 0 < x < \infty, \enspace t > 0,\\ & y(0,t) = f(t), \quad y(x,0) = 0 , \end{aligned}$$ where $f(t)$ is some function. Assume that $y$ is bounded. Give the answer as a convolution.