Linear Second Order ODE/y'' + y = 0/Proof 1

Proof
Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:


 * $p \dfrac {\d p} {\d y} = -y$

where $p = \dfrac {\d y} {\d x}$.

From:
 * First Order ODE: $y \rd y = k x \rd x$

with $k = 1$, this has the solution:


 * $p^2 = -y^2 + C$

or:
 * $p^2 + y^2 = C$

As the is the sum of squares, $C$ has to be positive for this to have any solutions.

Thus, let $C = \alpha^2$.

Then:

From Multiple of Sine plus Multiple of Cosine: Sine Form, this can be expressed as:


 * $y = C_1 \sin x + C_2 \cos x$