![]() ![]() For example, a=b=c=0 is a solution for any n, but such solutions are all obvious and uninteresting, and hence "trivial". If rank of 3 × 3 homogeneous matrix is less than 3 then the corresponding equations will have non-trivial solutions. Clearly, there are some solutions to the equation. Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation a^n b^n = c^n when n is greater than 2. In some cases, there may be other solutions ( sinusoids), which are called "nontrivial". This solution is considered obvious and is called the "trivial" solution. The differential equation f''(x)=-\lambda f(x) with boundary conditions f(0) = f(L) = 0 is important in math and physics, for example describing a particle in a box in quantum mechanics, or standing waves on a string. The trivial solution is y = 0, the zero function while a nontrivial solution is y ( x) = e x, the exponential function. For k 3 let A 1,N be a set not containing a solution to a1x1 . For example, consider the differential equation y'=y where y = f( x) is a function whose derivative is y′. Non-trivial solutions to a linear equation in integers. These solutions are called the trivial solutions. 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.
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