 Solution

Write the equation as $Ly=f$ with $L=-d^2/d{x}^2+1$. Homogeneous solutions vanishing at the left and right endpoints are $u_L(x)=\sinh (1+x)$ and $u_R(x)=\sinh (1-x)$. Their Wronskian is $u_L{u}_R^{\prime }-u_L^{\prime }{u}_R=-\sinh 2$. The Dirichlet Green kernel for minus second derivative plus one is thereforeIt vanishes at $x=\pm 1$, is continuous at $x=s$, and its first derivative has jump $-1$. Thus $LG(\cdot ,s)={\delta }_s$ in the distributional derivative sense, with the sign appropriate to $L=-d^2/d{x}^2+1$. Integration against $f$ gives the required boundary-value solution. The kernel is positive and symmetric, as is also apparent in the figure.

For the specified step forcing only $0\le s\le 1$ contributes at $x=0$, givingThe discontinuity in $f$ gives a piecewise twice-differentiable solution with continuous value and first derivative, rather than a jump in $y$ itself.