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Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinarydifferential equations taking the form \[u'' + u^2 = \alpha f(x) \] forparameter $\alpha$ are studied. The problem is defined on the $x$ line withdecay of both the solution $u(x)$ and the imposed forcing $f(x)$ as $|x| \to\infty $. The rate of decay of $f(x)$ is important and has a strong influenceon the structure of the solution space. Three particular forcings are examinedprimarily: a rectilinear top-hat, a Gaussian, and a Lorentzian, the latter twoexhibiting exponential and algebraic decay, respectively, for large $x$. Theproblem for the top hat can be solved exactly, but for the Gaussian and theLorentzian it must be computed numerically in general. Calculations suggestthat an infinite number of solution branches exist in each case. For thetop-hat and the Gaussian the solution branches terminate at a discrete set of$\alpha$ values starting from zero. A general asymptotic description of thesolutions near to a termination point is constructed that also providesinformation on the existence of local fold behaviour. The solution branches forthe Lorentzian forcing do not terminate in general. For large $\alpha$ theasymptotic analysis of Keeler, Binder $\&$ Blyth (2018 "On the criticalfree-surface flow over localised topography", J. Fluid Mech., 832, 73-96) isextended to describe the behaviour on any given solution branch using a methodfor glueing homoclinic connections.

Termination points and homoclinic glueing for a class of inhomogeneous nonlinear ordinary differential equations