Existence of Solution of 2nd Order Linear ODE

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

Let $$x_0 \in I$$, and let $$y_0 \in \R$$ and $$y_0' \in \R$$ be arbitrary.

Then the initial value problem:
 * $$\frac {d^2y}{dx^2} + P \left({x}\right) \frac {dy}{dx} + Q \left({x}\right) y = R \left({x}\right), y \left({x_0}\right) = y_0, y' \left({x_0}\right) = y_0'$$

has one and only one solution $$y = y \left({x}\right)$$ on the interval $$a \le x \le b$$.

Proof
Let us introduce the variable $$z = \frac {dy}{dx}$$.

Then the initial ODE can be written:
 * $$\begin{cases}

\dfrac {dy}{dx} = z & : y \left({x_0}\right) = y_0 \\ & \\ \dfrac {dz}{dx} = - P \left({x}\right) z - Q \left({x}\right) y + R \left({x}\right) & : z \left({x_0}\right) = y_0' \end{cases}$$

The converse is also true.

The result follows from Existence of Solution to System of First Order ODEs.