# A Linear Homogeneous Second-Order Differential Equation with Constant Coefficients

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This Demonstration shows how to solve a linear homogeneous differential equation with constant coefficients , where and are constant. First solve the characteristic equation . If and are two real roots of the characteristic equation, then the general solution of the differential equation is , where and are arbitrary constants. If , the general solution is . If , the general solution is .

Contributed by: Izidor Hafner (February 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.

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