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I have been trying to solve the following ODE and cannot get anywhere with it:

$\frac {d^{2}y}{dx^{2}} = 1-( \frac {dy}{dx})^{2}$.

The techniques I have used which are really the techniques for simple 2nd order ODEs don't work at all. Using the "D" operator doesn't work.

I am aware that I could write the RHS as $(1-dy/dx)(1+dy/dx)$ but it seems that doesn't get me very far. I am also aware I could try to solve this using infinite series, i.e $y = a_{0}+a_{1}x+... $ and do tedious calculations to get somewhat of an answer, but I am hoping for a closed form answer.

Any help would be appreciated.

(BTW this is not, strictly speaking, homework, just a puzzle I've been thinking about).

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Let $dy/dx=u$. Then the equation becomes $$ u'=1-u^2, $$ which is a first order ODE that can be solved by separation of variables.

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  • $\begingroup$ Wow. So simple. I feel stupid. $\endgroup$
    – JJG
    Jan 30, 2014 at 17:10

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