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

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

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


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

\dfrac {\d y} {\d x} = z &, \map y {x_0} = y_0 \\ & \\ \dfrac {\d z} {\d x} = -\map P x \dfrac {\d y} {\d x} - \map Q x y + \map R x &, \map z {x_0} = {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.