A Stability Analysis of Nonlinear Bessel-type ODE

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Is there an approach to the following 2nd order nonlinear ODE?
<br /> xy&#039;&#039; + 2 y&#039; = y^2 - k^2<br />
I am interested in learning how to analyze for asymptotic behavior, proof of existence, etc.
 
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... Anyone know how to do that?
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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