Existence of Solution of 2nd Order Linear ODE

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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:

$\displaystyle \frac {d^2y}{dx^2} + P \left({x}\right) \frac {dy}{dx} + Q \left({x}\right) y = R \left({x}\right), y \left({x_0}\right) = y_0, y' \left({x_0}\right) = y_0'$

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 $z = \dfrac {dy}{dx}$.

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.

$\blacksquare$