What does "trivial solution" mean? What does "trivial solution" mean exactly?
Must the trivial solution always be equal to the zero-solution (where all unknowns/variables are zero)?
 A: 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 $C = \infty$.
A: This is a quote from Surely You're Joking MR. Feynman.

A Different Box of Tools
At the Princeton graduate school, the physics department and the math department shared a
common lounge, and every day at four o'clock we would have tea. It was a way of relaxing in the
afternoon, in addition to imitating an English college. People would sit around playing Go, or
discussing theorems. In those days topology was the big thing.
I still remember a guy sitting on the couch, thinking very hard, and another guy standing in front
of him, saying, "And therefore such-and-such is true."
"Why is that?" the guy on the couch asks.
"It's trivial! It's trivial!" the standing guy says, and he rapidly reels off a series of logical steps:
"First you assume thus-and-so, then we have Kerchoff's this-and-that; then there's Waffenstoffer's
Theorem, and we substitute this and construct that. Now you put the vector which goes around here
and then thus-and-so . . ." The guy on the couch is struggling to understand all this stuff, which
goes on at high speed for about fifteen minutes!
Finally the standing guy comes out the other end, and the guy on the couch says, "Yeah, yeah.
It's trivial."
We physicists were laughing, trying to figure them out. We decided that "trivial" means
"proved." So we joked with the mathematicians: "We have a new theorem--that mathematicians can
prove only trivial theorems, because every theorem that's proved is trivial."
The mathematicians didn't like that theorem, and I teased them about it. I said there are never
any surprises-- that the mathematicians only prove things that are obvious.

A: "Trivial" means something to the effect of "obvious". Obvious depends on context. For example, the Riemann Zeta Function satisfies the equation
$$\zeta(s) = 2^s\pi^{s-1} \sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$$
and so it has obvious zeros for $s=-2,-4,-6,...$, and these are referred to as "trivial zeros". (Again, this depends on context. You have to know this functional equation to consider it "trivial")
In the context of differential equations, a trivial solution usually refers to the zero-function. However, I'm sure that some would refer to trivial solutions in cases when specific functions are obvious solutions.
See further information in the Triviality Wikipedia article.
A: The modifier "trivial" is always used in math in a single sense. It would not be wise to give it a definition here, so let me use some examples to give you a feeling of the sense. 
1) Let $a,b,c$ be not all $0$. Then $(0,0,0)$ is a trivial solution to the equation $ax+by+cz = 0$, because you can "see" it at once.
2) To prove that $z\bar{z} = |z|^{2}$ for every complex $z$, the case where $z=0$ is a trivial case.
3) The statement that every real number is a real number is a trivial statement.
From the above, you see that "trivial" in math much or less means "obvious" or "non-surprising". It perhaps sometimes connotes "boring"?
A: Depending on the context, trivial solutions can be:


*

*the zero function $y = 0$

*singular solutions of the differential equation (e.g. where you divide by $0$ in the process of solving the differential equation)

*constant functions $y = c, c \in \mathbb{R}$

*or just solutions which one can see immediately


However, I would use the the term 'trivial solution' for the zero function only, as this is the most common use of that term in mathematics (e.g. the center of that group is non- trivial (for p-groups), the solution set of that system of equations is trivial, etc.)
A: The etymology of the word "trivial " has to do with things that are commonly known.   Literally "three-way", like things that would be known by most people at a marketplace,  say, or anywhere where three paths come together. ..
Mike Spivak wrote that "when mathematics is done correctly it is trivial"  (in his book calculus on manifolds)...
