The equation for $y(x)$ is
$$ a=b(1-x^2)(1+y'^2)+y \qquad,\quad y(0)=0 $$
Where $a=y(1)$, $b >0$, and we may consider $0 \leq x \leq 1$. The solution is expected to be continuous but not necessarily differentiable at $x=0$.
Question I conjecture that there are no real, monotonically increasing solutions, but am unable to demonstrate this. How to do so? The conjecture may be incorrect.
I've played around with it, and asked Mathematica with no success. I've stared at the vector plot of $(1,y')$ in the $x-y$ plane. I've started to construct a perturbation series solution, but it's difficult to have faith in it viz. possible nonexistence of solutions.
Background The equation comes from this question on physics stackexchange. $y(x)$ is the inner surface of a bowl in which a particle slides without friction, such that (in my interpretation of the question) the $x$ projection of the motion is simple harmonic. The constant $a$ is the (conserved) energy, and $b=\omega^2/2$, where $\omega$ is the angular frequency of oscillation of $x(t)$.
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