Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions
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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:
- $\map {y'} {x_0} = {y_0}'$
Proof
Let $z = \dfrac {\d y} {\d x}$.
Then a solution to $(1)$ will yield a particular solution to:
- $(2): \quad \begin{cases} \dfrac {\d y} {\d x} = z & , \map y {x_0} = y_0 \\ & \\ \dfrac {\d z} {\d x} = -\map P x \dfrac {\d y} {\d x} - \map Q x y + \map R x & , \map z {x_0} = {y_0}' \end{cases}$
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
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.14$: Second Order Linear Equations: Introduction: Theorem $\text {A}$