Trivial Solution

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

• Example(s):
  • [math]\displaystyle{ y=0 }[/math] is a trivial solution to the equation [math]\displaystyle{ y'- y = 0 }[/math]
  • ...
• Counter-Example(s):
  • a Non-Trivial Solution, such as [math]\displaystyle{ y'=e^x }[/math] to the equation [math]\displaystyle{ y'- y = 0 }[/math]
• See: Equation, Problem, Solution, Trivial Group.

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References

2015

• (Wikipedia, 2015) ⇒ http://wikipedia.org/wiki/Triviality_(mathematics)#Trivial_and_nontrivial_solution
  • QUOTE: In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure.

Examples include:

• empty set: the set containing no members
• trivial group: the mathematical group containing only the identity element
• trivial ring: a ring defined on a singleton set.

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) = e^x, 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".

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