# A nonlinear second order differential equation.

Working on the geometry of submanifolds in a warped product, I came across the following nonlinear second order O.D.E.:

$\frac{d^2\lambda}{dt^2} = -(k_0\csc^2\theta)\lambda + (c\cdot\csc^2\theta)\lambda^2 - \frac{\cot^2\theta}{\lambda}\frac{d\lambda}{dt}$,

where $\lambda:\mathbb{R}\rightarrow\mathbb{R}_{>0}$ is a $C^\infty(\mathbb{R})$ function, and $k_0,\theta, c$ are constants. As you can see, the function:

$F(t,\lambda,\lambda'):= -K_0\lambda + C\lambda^2 - D\frac{\lambda'}{\lambda}$,

is continuous in all its arguments, which I think must suffice for the existence of its solutions. Nevertheless, I'm not so sure about its uniqueness, and the literature available to me at present is scarce about this point, and the behavior of its solution(s) for large (or "infinite") periods of time as well. Any help with these two topics or references whatsoever will be fully appreciated. Thanks.

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