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.