Details

The homogeneous linear differential equation

where is a function of , has a general solution of the form

,

where , , ..., are linearly independent particular solutions of the equation and , , …, are arbitrary constants.

If the coefficients , , …, are constant, then the particular solutions are found with the aid of the characteristic equation

.

To each real root of the characteristic equation of multiplicity , there corresponds particular solutions , , …, .

To each pair of imaginary roots of multiplicity , there corresponds pairs of particular solutions

, ,

, ,

…

, .

Reference

[1] V. P. Minorsky, Problems in Higher Mathematics, Moscow: Mir Publishers, 1975 p. 261.