Looking for a function in system of DE

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SUMMARY

The discussion focuses on deriving a function f that governs the behavior of two differential equations (DE) involving functions r_{1}(x) and r_{2}(x). The equations are defined as follows: \(\frac{d^{2}r_{2}(x)}{dx^{2}}=\frac{f(r_{2}(x)-r_{1}(x))}{C}\) and \(\frac{d^{2}r_{1}(x)}{dx^{2}}=-\frac{f(r_{2}(x)-r_{1}(x))}{K}\), with conditions that \(r_{2}(x)-r_{1}(x)>0\) and \(f(r_{2}(x)-r_{1}(x))>0\) for constants C and K greater than zero. The proposed function f is suggested to be a quadratic function, specifically \(3.31143 - 1.37286 x + 0.135714 x^2\), which approximately fits the lower curve of the system.

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I'm looking for prescription of function [itex]f[/itex] with behavior:
attachment.php?attachmentid=70294&stc=1&d=1401779566.png


putted in this system of DE solving some simple functions [itex]r_{1}(x)[/itex] and [itex]r_{2}(x)[/itex]:
[itex]\frac{d^{2}r_{2}(x)}{dx^{2}}=\frac{f(r_{2}(x)-r_{1}(x))}{C}[/itex]
[itex]\frac{d^{2}r_{1}(x)}{dx^{2}}=-\frac{f(r_{2}(x)-r_{1}(x))}{K}[/itex]

[itex]r_{2}(x)-r_{1}(x)>0;f(r_{2}(x)-r_{1}(x))>0;C,K>0[/itex]
 

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Why not a straight line?
 
3.31143 - 1.37286 x + 0.135714 x^2

fits more or less the lower curve
 

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