A linear homogeneous ordinary differential equation with constant coefficients has the general form of

where are all constants.

A second order linear homogeneous ordinary differential equation with constant coefficients can be expressed as

This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable , since constant coefficients are not capable of correcting any irregular formats or extra variables. An elementary function which satisfies this restriction is the exponential function .

Substitute the exponential function into the above differential equation, the characteristic equation of this differential equation is obtained

This characteristic equation has two roots and .

2nd Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic Equation ,		General Solution
1
2
3

Similar to the second order equations, the form, characteristic equation, and general solution of order linear homogeneous ordinary differential equations are summarized as follows:

nth Order Linear Homogeneous ODE with Constant Coefficients:
Characteristic Equation:
Solutions of Characteristic Equation		General Solution
1	are all different real numbers.
2	are k repeated real roots; others are different real numbers.
3	are k/2 pairs of complex conjugate roots ; others are different real numbers.