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How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach).

The numbers $g$ and $\ell$ are fixed constants and the initial conditions are $x(0)=x_0$ and $\dot x(0)=\omega_0$, where $x_0$ and $\omega_0$ are real constants.

According to my textbook on ODE it is possible, but since I am an engineer, I don't know how to formally prove this.

Can someone help me?

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The idea is to proof that any ODE which is of the form $$\dot{x}=f(x) \ \ \ \ \ \ x \in \mathbb{R}^n$$ $$x(t_0)=x_0$$ or even for a non-autonomous system (i.e. the right side depends explicitly on time), under some assumptions, (Lipschitz continuity of f with respect to x and perhaps something else(boundedness)). The idea is that the equivalent integral equation $$ x(t)=x_0+\int_{t_0}^{t_0+\delta} f(x(t))dt $$ can be written as a fixed point problem by takin $$\varphi(x(t))=x_0+\int_{t_0}^{t_0+\delta} f(x(t))dt $$ and prove that $\varphi^n$ is a contraction with the uniform norm, since the space of continuos functions is complete with that norm there must exist a solution to the original problem
for more detail you can check the book
Lectures on ordinary differential equations by J. Sotomayor (originally in portuguese)
Because your original problem fulfill those conditions, it must have a solution

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    $\begingroup$ It might help to mention that the theorem is called the Picard-Lindelof theorem. It is an easy theorem to look up online. $\endgroup$ – Ian Jan 12 '15 at 22:50

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