Laplace II: Steps, Impulses & Convolution · Superchallenge problem

A cantilever beam on $0\le x\le1$ (clamped at $x=0$, free at $x=1$) carries a unit point load at $x=\tfrac12$. Its deflection satisfies $y^{(4)}=\delta\!\left(x-\tfrac12\right)$ with $y(0)=y'(0)=0$ and $y^{\prime\prime}(1)=y^{\prime\prime\prime}(1)=0$. Use Laplace transforms in $x$ to find the tip deflection $y(1)$.