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$\begingroup$ Since the zero solution is the "obvious" solution, hence it is called a trivial solution. Any solution which has at least one component non-zero (thereby making it a non-obvious solution) is termed as a "non-trivial" solution. $\endgroup$ –
Trivial solution is a technical term. For example, for the homogeneous linear equation $7x+3y-10z=0$ it might be a trivial affair to find/verify that $(1,1,1)$ is a solution. But the term trivial solution is reserved exclusively for for the solution consisting of zero values for all the variables. There are similar trivial things in other topics.
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.)
Usually, the 0 0 -vector as a solution of a homogenous equation is called the "trivial solution". Any other solution is therefore "nontrivial". To expand on the comment of @Peter, the reason for using the word "trivial" in that context is that it takes a trivial amount of effort to verify that 0 0 is a solution of any homogeneous equation.
In my perspective, these trivial solutions are somehow referred to the asymptotic behavior of the equation but this expression is yet to be proven. sometimes the general solution is a function of these trivial solutions and sometimes these trivial solutions appear themselves as an addition in the general solution (or sometimes multiplication).
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a31x1 + a32x2 + a33x3 = 0; The homogeneous linear system has the two kinds of solutions. 1). zero solution. The sufficient and necessary condition of the zero solution for is listed as follows: det(A) ≠ 0; 2). Non-zero solution: The sufficient and necessary condition of the zero solution is listed as follow: det(A) = 0;
3 1. 1. I agree with @kdefaoite.The constant solution is trivial in the sense that it makes the differential character of the equation disappear (y = y−2 y = y − 2). So "a trivial solution is y = 1 y = 1 " seems fine. IMO, you could even say "this equation admits the trivial solution", where y = 0 y = 0 is implied. – user65203.
So the solution has at least one free variable. You can pick the value of the free variable as you please, specifically not $0$, and get a non-trivial solution. Share
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