Trivial Solution of Homogeneous Linear 2nd Order ODE

Theorem

The homogeneous linear second order ODE:

$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$

has the particular solution:

$\map y x = 0$

that is, the zero constant function.

This particular solution is referred to as the trivial solution.

Proof

We have:

$\map {\dfrac {\d} {\d x} } 0 = 0$

and so:

$\map {\dfrac {\d^2} {\d x^2} } 0 = 0$

from which:

$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$

Hence the result.

$\blacksquare$

Sources

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