Can this non-linear boundary-value problem be solved analytically?

Asked Apr 17 '23 at 18:03

Active Apr 24 '23 at 22:52

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I am trying to solve $$ [y^2(x)]''+\frac{x}{2} y'(x) = 0 $$ on $x\in\mathbb{R}$, with the conditions $$ \lim_{x\rightarrow-\infty} y(x) = a,\qquad \lim_{x\rightarrow\infty} y(x) = b. $$

I am particularly interested in non-negative solutions, $y(x)\geq0$, and the case where $a\neq b$. This means I cannot integrate by parts on the second term, so I'm rather stuck. Searching the usual places (Google, Wolfram alpha) hasn't returned anything interesting.

asked Apr 17 '23 at 18:03

ArrhythmicTiling

I can't promise you this works, but mybe try a power series approach... – Hyperbolic PDE friend Apr 17 '23 at 18:06
@Meowdog. The problem will be to handle the conditions at $\pm \infty$ – Claude Leibovici Apr 18 '23 at 06:20
Yes, but maybe start with any initial value first. Then you can adjust it accordingly – Hyperbolic PDE friend Apr 18 '23 at 06:36
1

Taking $y=x^2z$, $x=e^t$ then $z'_t=p(z)$ the equation is reduced by one order to an Abel equation of the second kind $zpp'_z+3zp+2z^2+(p+2z)^2+\frac14(p+2z)=0.$ – Eli Bartlett Apr 19 '23 at 05:35

Can this non-linear boundary-value problem be solved analytically?

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