Trivial Solution of Homogeneous Linear 2nd Order ODE

Theorem
The homogeneous linear second order ODE:
 * $\dfrac {\mathrm d^2 y} {\mathrm d x^2} + P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} + Q \left({x}\right) y = 0$

has the particular solution:
 * $y \left({x}\right) = 0$

that is, the zero constant function.

This particular solution is referred to as the trivial solution.

Proof
We have:
 * $\dfrac {\mathrm d} {\mathrm d x} \left({0}\right) = 0$

and so:
 * $\dfrac {\mathrm d^2} {\mathrm d x^2} \left({0}\right) = 0$

from which:
 * $\dfrac {\mathrm d^2 y} {\mathrm d x^2} + P \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x} + Q \left({x}\right) y = 0$

Hence the result.