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

Theorem
Let $\map P x$, $\map Q x$ and $\map R x$ be continuous real functions on a closed real interval $\closedint a b$.

Let $x_0$ be any point in $\closedint a b$.

Let $y_0$ and ${y_0}'$ be real numbers.

Then the linear second order ordinary differential equation:
 * $(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$

has a unique particular solution $\map y x$ on $\closedint a b$ such that:
 * $\map y {x_0} = y_0$

and:
 * $\map {y'} {x_0} = {y_0}'$