I can transform this equation into a more standard form: $$u^2 z_{uu} - u z_u + u^2 z = z^2.$$ I found a paper that kind of answers my question: http://www.sciencedirect.com/science/article/pii/0898122196000569
It would be nice to show that the solution is Lyapunov stable about z(u=0) = 0...
Multiple both sides by ##\frac{df}{dx}##. Then you get $$L^2 \frac{d^2f}{dx^2} \frac{df}{dx} = sinh(f) \frac{df}{dx}.$$ Notice that ##\frac{d^2f}{dx^2} \frac{df}{dx} = \frac{1}{2} \frac{d}{dx} (\frac{df}{dx})^2 ##. Integrate both sides with respect to x and you get $$L^2(\frac{df}{dx})^2 =...
You can let w = -y' and reduce the equation to an Abel Equation: ww'-w = f'/2, f = -ax^b Unfortunately, known closed form solutions of Abel's equations only exist in special situations, and this isn't one of the known ones.
Here is a paper on ways to approach Abel's equations...
I would tackle this problem using finite element methods. There is some work to set up the problem, but not technically out of reach of a doctoral student.
Is there an approach to the following 2nd order nonlinear ODE?
xy'' + 2 y' = y^2 - k^2
I am interested in learning how to analyze for asymptotic behavior, proof of existence, etc.