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

Definition

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

Also see

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

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.14$: Second Order Linear Equations: Introduction