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

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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.

$\blacksquare$


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