 Solution

For $y<0$, the Heaviside function vanishes, so $w=Ay+B$. For $y>0$, $w^{\prime \prime }=-1$, giving $w=-y^2/2+Cy+D$. Continuity of $w$ and $w^{\prime }$ at zero requires $D=B$ and $C=A$. Consequently the general continuously differentiable profile isIts second derivative jumps from zero to minus one. The equation is satisfied classically away from zero and in the weak, or distributional, sense across it; no globally twice-differentiable function can have the prescribed step as its second derivative. There is no Dirac delta contribution because $w^{\prime }$ is continuous. The value assigned to the Heaviside function at a single point does not change this weak solution.