Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

Theorem
Let $c_1$ and $c_2$ be real numbers.

Let $y_1 \left({x}\right)$ and $y_2 \left({x}\right)$ be particular solutions to the homogeneous linear second order ODE:
 * $(1): \quad \dfrac {\mathrm d^2 y} {\mathrm d x^2} + P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} + Q \left({x}\right) y = 0$

Then:
 * $c_1 y_1 \left({x}\right) + c_2 y_2 \left({x}\right)$

is also a particular solution to $(1)$.

That is, a linear combination of particular solutions to a homogeneous linear second order ODE is also a particular solution to that ODE.

Proof
Hence the result.