# Trivial Solution of Homogeneous Linear 1st Order ODE

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## Theorem

The homogeneous linear first order ODE:

- $\dfrac {\d y} {\d x} + \map Q x y = 0$

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:

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

from which:

- $\dfrac {\d y} {\d x} + \map Q x y = 0$

Hence the result.

$\blacksquare$