While testing Physics-informed neural networks I discovered that IVP
$$ y'(x) = x^2 - y^2 $$ $$ y(0) = 2.5 $$
possibly has a second solution. I'm quite sure that my model once found a very good approximation (I discarded it because I wasn't looking for it) of this "new" solution. But I cannot recreate it since, so I started wondering whether this IVP actually has a second solution.
My best attempt so far at finding the second solution is shown on this graph as a blue curve.
Curves on the graph are:
- Blue curve - best neural network solution
- Red curve - known solution found with a RK45 numerical method
- Green curve - actual derivative of NN solution
- Orange curve - value of $f(x, y) = x^2 - y^2$ computed from the NN solution
Is it possible to check if that potential second solution actually exists or does not exist, without finding it?
I know that this IVP is a form of Riccati equation and its known solution has a (not simple) analytical form. So if the potential second solution actually exists, I am wondering if it has maybe been already found or the existing analytical solution can be adapted to fit the potential "second" solution.
$f(x,y)=x^2-y^2$ and $\frac{\partial f}{\partial y}=-2y$ are continuous functions of $x$ and $y$ in the region $-a<x<a$ and $2.5-b<y<2.5+b$ for any positive real numbers $a$ and $b$. Due to the existence and uniqueness theorem for 1st order ODEs (See e.g. S. L. Ross), there is a unique solution of the IVP $y'=x^2-y^2$, $y'(0)=2.5$.