Zero function is often a solution?

Question

I wonder why do we get zero as the trivial solution many a times. For instace:

1. The ordinary differential equation $y'=y(y-1)$ has $y=0$ and $y=1$ as its trivial solutions.
2. Zero is the trivial solution of the homogenous system of line eqations.

Why zero is the trivial solution for many problems in mathematics? Please explain . Thanks!

Answer 1

The zero solution is called ‘trivial’ because it is the most obvious and straightforward one. If you ask why it comes up most of the time, that’s probably how $0$ works.

In the first case, you can see that if we make the right side equal to zero by plugging in $y=0,1$ then we would have $y’=0$ suggesting that $y$ is a constant function. Normally, one would not be looking for constant function solutions so these are termed trivial.

Any system of the form: $$a_1 x + b_1 y +c_1 z ...=0$$ $$a_2 x + b_2 y +c_2 z ...=0 \\ . \\ . \\. $$ has a trivial solution when all the variables are zero.

Answer 2

Very often statements about zero objects are vacuously true - i.e. the converse simply isn't true. This happens because attributing some property to a zero object (zero vector of some vector space, the zero function or the empty set, to name a few), which represents the idea of no object at all, is quite silly if not impossible.

As to why we refer to such solutions as "trivial solutions", it is because many of these solutions are very easy to find, and are quite dull, so finding that solution or saying that it exists doesn't grant one a deeper understanding of the problem at hand.

Asking mathematical questions (at every level) can lead one into a blessed rabbit hole - given one has some passion for problem solving, or math, or both - and the beauty or usefulness of the answer - maybe even the sheer scope of new questions and directions one can go to afterwards - will not reveal themselves from looking at the solution which one can obtain "for free", which as we've seen is often the zero solution.