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\[ y\ddot{y}-\dot{y}^2=1 \] Which method would I use here? The boundary conditions are \[y(a)=y(-a)=1 \]

Mathematics
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i think reduction of order with\[p=\dot y\]will lead us to something maybe?
just note that\[\ddot y=p\dot p \]
\[ \dot{p}=\frac{ dp }{ dx}= \frac{d^2y}{dx^2}=\ddot{y} \]Surely?

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Other answers:

[yy']'=yy''+y'^2 so just difference in sign. Maybe it is a derivative of some quotient.
make p a function of y\[\ddot y=\frac{dp}{dx}=\frac{dp}{dy}\frac{dy}{dx}=p \dot p\] and what @myko mentioned can be a good start too
actually that is\[\ddot y=\frac{dp}{dx}=\frac{dp}{dy}\frac{dy}{dx}=p \dot p_y\]
Sorry, I got confused by the notation
\[\Large yy'' - \left( y' \right)^2 = -\left(y'\right)^2\left( \frac{y}{y'} \right)'\]
Very nice, but how does that help?

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