What is a trivial solution?

In mathematics, a trivial solution is one that is considered to be very simple and poses little interest for the mathematician. Typical examples are solutions with the value 0 or the empty set, which does not contain any elements.

The equation x + 5y = 0 contains an infinity of solutions. Among these, the solution x = 0, y = 0 is considered to be trivial, as it is easy to infer without any additional calculation. All other solutions are nontrivial. Similarly, the differential equation y' = y has the trivial solution y = 0 and the nontrivial solution y(x) = exp(x). The latter is the exponential function, which raises the natural base e = 2.7182818 to the power of x.