Let $\widehat{y}(p)$ be the Laplace transform of $y(t)$, where $y(t)$ satisfies

$$y^{\prime}(t)=y(\pi-t)$$

and

$$y(0)=1 ; \quad y(\pi)=k ; \quad y(t)=0 \text { for } t<0 \text { and for } t>\pi$$

Show that

$$p \widehat{y}(p)+k e^{-\pi p}-1=e^{-\pi p} \widehat{y}(-p)$$

and hence deduce that

$$\widehat{y}(p)=\frac{(k+p)-(1+p k) e^{-\pi p}}{1+p^{2}}$$

Use the inversion formula for Laplace transforms to find $y(t)$ for $t>\pi$ and deduce that a solution of the above boundary value problem exists only if $k=-1$. Hence find $y(t)$ for $0 \leqslant t \leqslant \pi$.