[Math] What does “trivial solution” mean

Question

What does "trivial solution" mean exactly?

Must the trivial solution always be equal to the zero-solution (where all unknowns/variables are zero)?

Answer 1

It is not always the zero solution, but it always reflects solutions that one can "see" without actually having to solve anything, and for practical purposes they are almost always seen as irrelevant. They are also almost always "simpler" than the general solutions, and some times they cannot be expressed as part of a general solution formula.

For instance, a logistical system like, say, $y' = y(1-y)$ has two trivial solutions: $y(x) = 0$ and $y(x) = 1$ (trivial because they clearly make both sides of the equation equal $0$, and if you're looking for solutions like that, you easily find them without and calculations). The general solution, $y(x) = \frac{e^x}{C + e^x}$, can encompass one trivial solution ($y(x) = 1$, with $C = 0$), but it cannot encompass the other, since we're not allowed to put $C = \infty$.