Nonlinear second order inhomogeneous differential equations in one dimension

We study inhomogeneous nonlinear second-order differential equations in one dimension. The inhomogeneities can be point sources or continuous source distributions. We consider second order differential equations of type ${\varphi }^{\mathrm{\prime }\mathrm{\prime }}(x)+V(\varphi (x))=Q\delta (x)$, where $V(\varphi )$ is a continuous, differentiable, analytic function and $Q\delta (x)$ is a point source. In particular we study cubic functions of the form $V(\phi(x)) = A,\phi(x) + B,\phi^3(x)$. We show that Green functions can be determined for modifications of such cubic equations, and that such Green’s functions can be used to determine the solutions for cases where the point source is replaced by a continuous source distribution.