Trivial Solution

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A Trivial Solution is a solution to an equation or problem that has a simple structure.



References

2015

Examples include:

Trivial can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the differential equation

[math]\displaystyle{ y'=y }[/math]
where y = f(x) is a function whose derivative is y′. The trivial solution is
y = 0, the zero function
while a nontrivial solution is
y (x) = ex, the exponential function.
The differential equation [math]\displaystyle{ f''(x)=-\lambda f(x) }[/math] with boundary conditions [math]\displaystyle{ f(0) = f(L) = 0 }[/math] is important in math and physics, for example describing a particle in a box in quantum mechanics, or standing waves on a string. It always has the solution [math]\displaystyle{ f(x) = 0 }[/math]. This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial".
Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation [math]\displaystyle{ a^n + b^n = c^n }[/math] when n is greater than 2. Clearly, there are some solutions to the equation. For example, [math]\displaystyle{ a=b=c=0 }[/math] is a solution for any n, but such solutions are all obvious and uninteresting, and hence "trivial".