Existence of Solution to System of First Order ODEs

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Theorem

Consider the system of initial value problems:

$\begin{cases} \dfrac {\mathrm d y} {\mathrm d x} = f \left({x, y, z}\right) & : y \left({x_0}\right) = y_0 \\ & \\ \dfrac {\mathrm d z} {\mathrm d x} = g \left({x, y, z}\right) & : z \left({x_0}\right) = z_0 \\ \end{cases}$

where $f \left({x, y, z}\right)$ and $g \left({x, y, z}\right)$ are continuous real functions in some region of space $x y z$ that contains the point $\left({x_0, y_0, z_0}\right)$.

Then this system of equations has a unique solution which exists on some interval $\left|{x - x_0}\right| \le h$.


Proof