# 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) – Mo_ 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

I'm not going to answer this question in complete detail, since it might be homework, but here are some tips on how to proceed.

First, decide if there's an exact solution available. If it's linear, you have a good chance. The class of nonlinear ODE for which there is an exact solution can seem disappointingly small, but it's worth knowing the tricks (maybe it's exact, maybe you can use the Frobenius method, perhaps there is a change of variables that simplifies things, etc.). If stumped, another tool is a Computer Algebra System.

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.

• The Author - Thx for input. More info. y=1 is the dimensionless equilibrium position. Imagine lifting the meniscus up to say, y=2, initial velocity dy/dt = 0. The force of gravity initially pushes it down. But as the meniscus falls, its mass becomes smaller and a constant surface tension (pushing up) becomes greater than the gravity. So the meniscus slows down, reverses direction and goes back up. So it oscillates back and forth, non-linearly. Substitution v = dy/dt will get v as a function of y. Would like to get v as function of t non-numerically. BTW - this is not homework :) – user28321 Aug 18 '13 at 13:20