Linear ordinary differential equation with constant coefficients

An ordinary differential equation (cf. Differential equation, ordinary) of the form

(1)

where is the unknown function, are given real numbers and is a given real function.

The homogeneous equation corresponding to (1),

(2)

can be integrated as follows. Let be all the distinct roots of the characteristic equation

(3)

with multiplicities , respectively, . Then the functions

(4)

are linearly independent (generally speaking, complex) solutions of (2), that is, they form a fundamental system of solutions. The general solution of (2) is a linear combination, with arbitrary constant coefficients, of the fundamental system of solutions. If is a complex number, then for every integer , , the real part and the imaginary part of the complex solution are linearly independent real solutions of (2), and to a pair of complex conjugate roots of multiplicity correspond linearly independent real solutions

The inhomogeneous equation (1) can be integrated by the method of variation of constants. If is a quasi-polynomial, i.e.

where and are polynomials of degree , and if the number is not a root of (3), one looks for a particular solution of (1) in the form

(5)

Here and are polynomials of degree with undetermined coefficients, which are found by substituting (5) into (1). If is a root of (3) of multiplicity , then one looks for a particular solution of (1) in the form

by the method of undetermined coefficients. If is a particular solution of the inhomogeneous equation (1) and is a fundamental system of solutions of the corresponding homogeneous equation (2), then the general solution of (1) is given by the formula

where are arbitrary constants.

A homogeneous system of linear differential equations of order ,

(6)

where is the unknown vector and is a constant real matrix, can be integrated as follows. If is a real eigen value of multiplicity of the matrix , then one looks for a solution corresponding to in the form

(7)

Here are polynomials of degree with undetermined coefficients, which are found by substituting (7) into (6); there are exactly linearly independent solutions of the form (7). If is a complex eigen value of multiplicity , then the real and imaginary parts of the complex solutions of the form (7) form linearly independent real solutions of (6), and a pair of complex conjugate eigen values and of multiplicity of the matrix generates linearly independent real solutions of (6). Taking all eigen values of , one finds linearly independent solutions, that is, a fundamental system of solutions. The general solution of (6) is a linear combination, with arbitrary constant coefficients, of the solutions that form the fundamental system.

The matrix is the fundamental matrix of the system (7), normalized at the origin, since , the unit matrix. Here

and this matrix series converges absolutely for any matrix and all real . Every other fundamental matrix of the system (6) has the form , where is a constant non-singular matrix of order .

References

[1]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[2]	V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
[3]	B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)