Characteristic Polynomial

Not to be confused with the characteristic polynomial, a linear algebra concept.

When working with homogeneous systems of linear differential equations the characteristic polynomial (sometimes called the characteristic equation) is found by replacing the derivative order with a power.

Where is a stand-in variable. One finds that the roots of the equation are:

See Wikipedia, Undetermined Coefficients - Paul's Online Notes, Stack Exchange, and W2L6 - Homogeneous Linear ODEs for source, more information, and examples.