How to solve this ordinary differential equation?

Here is partial answer. I hope this question doesn't get closed and someone can give a better answer.

I looked at the numerical solution to the initial value problem $$\frac{dy}{dx} = 1 + \sqrt{1-xy}, y(0) = 0$$ the solution exist for only finite time.

The solution must satisfy $$x < y < 2x$$ and these line cut the boundary $xy = 1$ at $(\pm 1, \pm 1), (\pm \frac{1}{\sqrt 2}, \pm \sqrt 2)$ so the interval of existence cannot be bigger than $(-1, 1)$ and no smaller than $(-1/\sqrt 2, 1/\sqrt 2).$

The Runge-Kutta numerical solution I tried stops at $x = 0.75128$.

Tags:

Ordinary Differential Equations

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