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Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$:

$$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$

Let $f_1 (x)$ and $f_2 (x)$ be two known linearly independent solutions of the homogeneous system. Now consider adding a small constant inhomogeneity $c<<1$ to the differential equation:

$$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=c$$

Knowing $f_1 (x)$ and $f_2 (x)$, is there any procedure to approximate the solution of the inhomogeneous system using the fact that $c<<1$, without having to actually solve it?

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If you know $f_1(x)$ and $f_2(x)$, then you can easily construct a Green function for this linear ODE, with your favorite boundary condition. Solving an inhomogeneous system then comes down to integrating the inhomogeneity against the Green function, as usual. –  Igor Khavkine Apr 29 '13 at 12:49
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