# Non-linear ordinary differential equation in the modeling of the oscillation of a meniscus

I am trying to model the oscillation of a fluid miniscus in a straw when the miniscus is displaced from its equilibrium level. The results was the following non-linear ODE:

$$y''= 1/y - 1.$$

This can be solved numerically but I have not been able to find anyone who has worked on it. Is anyone familiar with this equation?

• Would math.stackexchange.com be a better home for this question? – Qmechanic Aug 16 '13 at 14:14
• @KyleKanos: Actually, I agree with that. – Qmechanic Aug 16 '13 at 14:29
• I feel that it might be solvable, you might ask people in math chatroom whether they recognize it. – unsym Aug 16 '13 at 15:01
• Use $p=y'$ and replace $y''=p\dfrac{dp}{dy}$. (The final integral that gives $y$ seems hard to calculate) – Mostafa Aug 16 '13 at 15:20
• This question appears to be off-topic because it is about undergraduate level mathematics (should be in math.stackexchange.com) rather than computational science. – Brian Borchers Oct 18 '13 at 17:34

If you decide there isn't an exact solution, you can choose a numerical method to compute an approximate solution. This depends on the boundary/initial conditions. Whether you have an "initial value problem" (information about $y$ and its derivatives at a single point in the domain) or a "boundary value problem" (information about $y$ and its derivatives at two separate points in the domain) makes a lot of difference.
Based on the comment below, you are interested in an initial value problem, which for concreteness we'll say is $y''(t) = 1/y - 1$, $y(0) = y_0$, $y'(0) = 0$. Now one can choose freely from the bestiary of ODE solvers available. Note that if you can show that the solution stays bounded away from $y=0$, things will behave nicely. The form of this second order ODE makes it possible to use second-order methods like Verlet methods, you can pick an old workhorse like Runge-Kutta-Fehlberg(aka ode45 from MATLAB), or, if it seems likely that the solution will approach $y=0$, an implicit and/or stiff solver like a . Note that by introducing a new variable $v \doteq dy/dt$, you can write the second order ODE as a system of two first order ODEs.