Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

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

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

Then:
 * $c_1 \, \map {y_1} x + c_2 \, \map {y_2} x$

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.