Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

Definition
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 known as a '''linear combination of particular solutions to $(1)$.

Also see

 * Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE