Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions/Proof

Proof
Let $z = \dfrac {\mathrm d y} {\mathrm d x}$.

Then a solution to $(1)$ will yield a solution to:


 * $(2): \quad \begin{cases}

\dfrac {\mathrm d y} {\mathrm d x} = z &, y \left({x_0}\right) = y_0 \\ & \\ \dfrac {\mathrm d z} {\mathrm d x} = -P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} - Q \left({x}\right) y + R \left({x}\right) &, z \left({x_0}\right) = {y_0}' \end{cases}$

From Lipschitz Condition on Linear ODE of Continuous Functions, $(2)$ satisfies the Lipschitz condition.

Hence Picard's Existence Theorem applies.

Hence the result.