View source for Linear ordinary differential equation with constant coefficients

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593702.png" /> is the unknown function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593703.png" /> are given real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593704.png" /> is a given real function.

The homogeneous equation corresponding to (1),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

can be integrated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593706.png" /> be all the distinct roots of the characteristic equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593707.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

with multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593708.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l0593709.png" />. Then the functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

are linearly independent (generally speaking, complex) solutions of (2), that is, they form a [[Fundamental system of solutions|fundamental system of solutions]]. The general solution of (2) is a linear combination, with arbitrary constant coefficients, of the fundamental system of solutions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937011.png" /> is a complex number, then for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937013.png" />, the real part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937014.png" /> and the imaginary part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937015.png" /> of the complex solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937016.png" /> are linearly independent real solutions of (2), and to a pair of complex conjugate roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937017.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937018.png" /> correspond <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937019.png" /> linearly independent real solutions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937020.png" /></td> </tr></table>

The inhomogeneous equation (1) can be integrated by the method of [[Variation of constants|variation of constants]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937021.png" /> is a quasi-polynomial, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937024.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937025.png" />, and if the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937026.png" /> is not a root of (3), one looks for a particular solution of (1) in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937029.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937030.png" /> with undetermined coefficients, which are found by substituting (5) into (1). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937031.png" /> is a root of (3) of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937032.png" />, then one looks for a particular solution of (1) in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937033.png" /></td> </tr></table>

by the method of undetermined coefficients. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937034.png" /> is a particular solution of the inhomogeneous equation (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937035.png" /> is a fundamental system of solutions of the corresponding homogeneous equation (2), then the general solution of (1) is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937036.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937037.png" /> are arbitrary constants.

A homogeneous system of linear differential equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937038.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937040.png" /> is the unknown vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937041.png" /> is a constant real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937042.png" /> matrix, can be integrated as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937043.png" /> is a real eigen value of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937044.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937045.png" />, then one looks for a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937046.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937047.png" /> in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937049.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937050.png" /> with undetermined coefficients, which are found by substituting (7) into (6); there are exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937051.png" /> linearly independent solutions of the form (7). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937052.png" /> is a complex eigen value of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937053.png" />, then the real and imaginary parts of the complex solutions of the form (7) form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937054.png" /> linearly independent real solutions of (6), and a pair of complex conjugate eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937056.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937057.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937058.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937059.png" /> linearly independent real solutions of (6). Taking all eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937060.png" />, one finds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937061.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937062.png" /> is the [[Fundamental matrix|fundamental matrix]] of the system (7), normalized at the origin, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937063.png" />, the unit matrix. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937064.png" /></td> </tr></table>

and this matrix series converges absolutely for any matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937065.png" /> and all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937066.png" />. Every other fundamental matrix of the system (6) has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937068.png" /> is a constant non-singular matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059370/l05937069.png" />.

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