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.
$\blacksquare$