Non linear 2nd order differential equation

25 July 2009

I am not able to find the general function that integrates the following 2nd order differential equation;

*y*^2 *y’’ + a y^*3 – *b = *0

in which 0 ≤ *a *≤ 1 is a constant;

*b *__>__ 0 is a constant.

**The equation relates to the behaviour (orbit/motion) of a massless point subjected to a particular central motion.**

I could easily find only the solutions associated with nil values for constants *a* and *b.*

*I would be deeply grateful to any friend who can help me resolve the problem.*

*Many thanks for any help.*

*Leo Klem*

Non linear 2nd order equation.Thanks, with a clarification

Many thanks to my two first interlocutors. I have to point out that the problem described by **the differential equation regards a central motion**. The integrating function, if it does exist, is actually useful in the explicit form given by

*y = f(x; a, b, C, D)*

in which* x *is the independent variable (it represent the "orbital" angle), *a *and *b* are the given equation constants, and* C, D *are the two integration constants.

Solutions in the form of an expression of functions of *y* do actually create difficult problems of interpretation as to the kind of motion regarded.

What beyond the inverse function?

Thanks to Chisigma. You got the correct procedure, but the solution is still away. It's the point where I had actually to stop. It's clearly a "solution" in the form of an inverse function, i.e., the argument expressed as a function of the dependent variable. The integral you indicate seems however difficult to solve, but in a few particular cases, such as - for instance - assuming *C1* = 0.

Could you please provide further suggestions?

*Nome a parte, sono italiano anch'io, ma non sono un matematico. Grato comunque.(Hi)*