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

Theorem
Let $P \left({x}\right)$, $Q \left({x}\right)$ and $R \left({x}\right)$ be continuous real functions on a closed real interval $\left[{a \,.\,.\, b}\right]$.

Let $x_0$ be any point in $\left[{a \,.\,.\, b}\right]$.

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

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

has a unique solution $y \left({x}\right)$ on $\left[{a \,.\,.\, b}\right]$ such that:
 * $y \left({x_0}\right) = y_0$

and:
 * $y' \left({x_0}\right) = {y_0}'$