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

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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 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:

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


Proof

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


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

$(2): \quad \begin{cases} \dfrac {\mathrm d y} {\mathrm d x} = z & , y \left({x_0}\right) = y_0 \\ & \\ \dfrac {\mathrm d z} {\mathrm d x} = -P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} - Q \left({x}\right) y + R \left({x}\right) & , z \left({x_0}\right) = {y_0}' \end{cases}$


From Lipschitz Condition on Linear ODE of Continuous Functions, $(2)$ satisfies the Lipschitz condition.

Hence Picard's Existence Theorem applies.

Hence the result.

$\blacksquare$


Sources