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Can someone point me in a direction to solve this kind of integral constrained system of ODEs.

\begin{align} &\int_0^{1/2}\dot{y}^2(t)=p\\ &2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\ &y(0)=0,y(1/2)=1/2 \end{align}

I have reduced it to 1st order: \begin{align} &\int_0^{1/2}x^2(t)=p\\ &\dot{y}=x \\ &2\lambda_1\dot{x}(t)+\pi cos(\pi y(t))=0\\ &y(0)=0,y(1/2)=1/2 \end{align}

but its still not suitable for an ODE solver. Any help will be appreciated.

Edit:$p$ is a known constant and $\lambda_1$ is an unknown constant.
Disclaimer: This is a cross-post and has some good answers here on MathSE.

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    $\begingroup$ If you introduce another dependent variable $\dot u=\dot y^2, u(0)=0, u(\frac12)=p$, then this is a BVP in a standard form, solvable by any good BVP solver (gist.github.com/ikirill/cd286cd22e73a343f97dcd1145dc958c). But it's not converging for me, I'm not sure why. $\endgroup$ – Kirill Dec 16 '18 at 17:24
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    $\begingroup$ See my answer to the cross-post math.stackexchange.com/a/3043031/115115, one needs to be very careful with the initial data to get convergence in the solver. This also does not tell if there are higher modes for this non-linear problem. $\endgroup$ – LutzL Dec 16 '18 at 20:07

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