Namespaces
Variants
Actions

Difference between revisions of "Linear ordinary differential equation with constant coefficients"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
l0593701.png
 +
$#A+1 = 69 n = 0
 +
$#C+1 = 69 : ~/encyclopedia/old_files/data/L059/L.0509370 Linear ordinary differential equation with constant coefficients
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
An ordinary differential equation (cf. [[Differential equation, ordinary|Differential equation, ordinary]]) of the form
 
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>
+
$$ \tag{1 }
 +
x  ^ {(} n) + a _ {1} x  ^ {(} n- 1) + \dots + a _ {n} x  = f ( t) ,
 +
$$
  
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.
+
where $  x ( t) $
 +
is the unknown function, $  a _ {1} \dots a _ {n} $
 +
are given real numbers and $  f ( t) $
 +
is a given real function.
  
 
The homogeneous equation corresponding to (1),
 
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>
+
$$ \tag{2 }
 +
x  ^ {(} n) + a _ {1} x  ^ {(} n- 1) + \dots + a _ {n} x  = 0 ,
 +
$$
 +
 
 +
can be integrated as follows. Let  $  \lambda _ {1} \dots \lambda _ {k} $
 +
be all the distinct roots of the characteristic equation
  
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
+
$$ \tag{3 }
 +
\lambda  ^ {n} + a _ {1} \lambda  ^ {n-} 1 + \dots + a _ {n-} 1 \lambda + a _ {n}  = 0
 +
$$
  
<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  $  l _ {1} \dots l _ {k} $,
 +
respectively,  $  l _ {1} + \dots + l _ {k} = n $.  
 +
Then the functions
  
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
+
$$ \tag{4 }
 +
e ^ {\lambda _ {j} t } ,\
 +
t e ^ {\lambda _ {j} t } \dots t ^ {l _ {j} - 1 } e ^ {\lambda _ {j} t } ,\  j = 1 \dots k ,
 +
$$
  
<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  $  \lambda _ {j} = \alpha _ {j} + \beta _ {j} i $
 +
is a complex number, then for every integer  $  m $,
 +
0 \leq  m \leq  l _ {j} - 1 $,
 +
the real part  $  t  ^ {m} e ^ {\alpha _ {j} t } \cos  \beta _ {j} t $
 +
and the imaginary part  $  t  ^ {m} e ^ {\alpha _ {j} t } \sin  \beta _ {j} t $
 +
of the complex solution  $  t  ^ {m} e ^ {\lambda _ {j} t } $
 +
are linearly independent real solutions of (2), and to a pair of complex conjugate roots  $  \alpha _ {j} \pm  \beta _ {j} i $
 +
of multiplicity  $  l _ {j} $
 +
correspond  $  2 l _ {j} $
 +
linearly independent real solutions
  
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
+
$$
 +
t  ^ {m} e ^ {\alpha _ {j} t } \cos  \beta _ {j} \ \
 +
\textrm{ and } \  t  ^ {m} e ^ {\alpha _ {j} t } \sin \
 +
\beta _ {j} t ,\  m = 0 \dots l _ {j} - 1 .
 +
$$
  
<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/l05937020.png" /></td> </tr></table>
+
The inhomogeneous equation (1) can be integrated by the method of [[Variation of constants|variation of constants]]. If  $  f $
 +
is a quasi-polynomial, i.e.
  
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.
+
$$
 +
f( t) = e  ^ {at} ( p _ {m} ( t)  \cos  bt + q _ {m} ( t)  \sin  bt ),
 +
$$
  
<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  $  p _ {m} $
 +
and  $  q _ {m} $
 +
are polynomials of degree  $  \leq  m $,
 +
and if the number  $  a + b i $
 +
is not a root of (3), one looks for a particular solution of (1) in the form
  
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
+
$$ \tag{5 }
 +
x _ {0} ( t)  = e  ^ {at} ( P _ {m} ( t) \cos  b t + Q _ {m} ( t) \sin  b t ) .
 +
$$
  
<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  $  P _ {m} $
 +
and  $  Q _ {m} $
 +
are polynomials of degree  $  m $
 +
with undetermined coefficients, which are found by substituting (5) into (1). If  $  a + b i $
 +
is a root of (3) of multiplicity  $  k $,
 +
then one looks for a particular solution of (1) in the form
  
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
+
$$
 +
x _ {0= t  ^ {k} e  ^ {at} ( P _ {m} ( t) \cos  b t + Q _ {m} ( t) \sin  b t )
 +
$$
  
<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  $  x _ {0} ( t) $
 +
is a particular solution of the inhomogeneous equation (1) and  $  x _ {1} ( t) \dots x _ {n} ( t) $
 +
is a fundamental system of solutions of the corresponding homogeneous equation (2), then the general solution of (1) is given by the formula
  
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
+
$$
 +
x( t)  = x _ {0} ( t) + C _ {1} x _ {1} ( t) + \dots + C _ {n} x _ {n} ( t) ,
 +
$$
  
<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  $  C _ {1} \dots C _ {n} $
 +
are arbitrary constants.
  
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  $  n $,
  
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" />,
+
$$ \tag{6 }
 +
\dot{x}  =  A x ,
 +
$$
  
<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  $  x \in \mathbf R  ^ {n} $
 +
is the unknown vector and  $  A $
 +
is a constant real  $  n \times n $
 +
matrix, can be integrated as follows. If  $  \lambda $
 +
is a real eigen value of multiplicity  $  k $
 +
of the matrix  $  A $,
 +
then one looks for a solution  $  x = ( x _ {1} \dots x _ {n} ) $
 +
corresponding to  $  \lambda $
 +
in the form
  
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
+
$$ \tag{7 }
 +
x _ {1}  = P _ {1} ( t) e ^ {\lambda t } \dots
 +
x _ {n}  = P _ {n} ( t) e ^ {\lambda t } .
 +
$$
  
<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  $  P _ {1} ( t) \dots P _ {n} ( t) $
 +
are polynomials of degree  $  k - 1 $
 +
with undetermined coefficients, which are found by substituting (7) into (6); there are exactly  $  k $
 +
linearly independent solutions of the form (7). If  $  \lambda $
 +
is a complex eigen value of multiplicity  $  k $,
 +
then the real and imaginary parts of the complex solutions of the form (7) form  $  2 k $
 +
linearly independent real solutions of (6), and a pair of complex conjugate eigen values  $  \lambda $
 +
and  $  \overline \lambda \; $
 +
of multiplicity  $  k $
 +
of the matrix  $  A $
 +
generates  $  2 k $
 +
linearly independent real solutions of (6). Taking all eigen values of  $  A $,
 +
one finds  $  2 n $
 +
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.
  
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  $  X ( t) = e  ^ {At} $
 +
is the [[Fundamental matrix|fundamental matrix]] of the system (7), normalized at the origin, since  $  X ( 0) = E $,  
 +
the unit matrix. Here
  
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
+
$$
 +
e  ^ {At}  = E +
 +
\sum _ { k= } 1 ^  \infty 
  
<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>
+
\frac{A  ^ {k} t  ^ {k} }{k ! }
 +
,
 +
$$
  
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" />.
+
and this matrix series converges absolutely for any matrix $  A $
 +
and all real $  t $.  
 +
Every other fundamental matrix of the system (6) has the form $  e  ^ {At} C $,  
 +
where $  C $
 +
is a constant non-singular matrix of order $  n $.
  
 
====References====
 
====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>
 
<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>

Revision as of 22:17, 5 June 2020


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

$$ \tag{1 } x ^ {(} n) + a _ {1} x ^ {(} n- 1) + \dots + a _ {n} x = f ( t) , $$

where $ x ( t) $ is the unknown function, $ a _ {1} \dots a _ {n} $ are given real numbers and $ f ( t) $ is a given real function.

The homogeneous equation corresponding to (1),

$$ \tag{2 } x ^ {(} n) + a _ {1} x ^ {(} n- 1) + \dots + a _ {n} x = 0 , $$

can be integrated as follows. Let $ \lambda _ {1} \dots \lambda _ {k} $ be all the distinct roots of the characteristic equation

$$ \tag{3 } \lambda ^ {n} + a _ {1} \lambda ^ {n-} 1 + \dots + a _ {n-} 1 \lambda + a _ {n} = 0 $$

with multiplicities $ l _ {1} \dots l _ {k} $, respectively, $ l _ {1} + \dots + l _ {k} = n $. Then the functions

$$ \tag{4 } e ^ {\lambda _ {j} t } ,\ t e ^ {\lambda _ {j} t } \dots t ^ {l _ {j} - 1 } e ^ {\lambda _ {j} t } ,\ j = 1 \dots k , $$

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 $ \lambda _ {j} = \alpha _ {j} + \beta _ {j} i $ is a complex number, then for every integer $ m $, $ 0 \leq m \leq l _ {j} - 1 $, the real part $ t ^ {m} e ^ {\alpha _ {j} t } \cos \beta _ {j} t $ and the imaginary part $ t ^ {m} e ^ {\alpha _ {j} t } \sin \beta _ {j} t $ of the complex solution $ t ^ {m} e ^ {\lambda _ {j} t } $ are linearly independent real solutions of (2), and to a pair of complex conjugate roots $ \alpha _ {j} \pm \beta _ {j} i $ of multiplicity $ l _ {j} $ correspond $ 2 l _ {j} $ linearly independent real solutions

$$ t ^ {m} e ^ {\alpha _ {j} t } \cos \beta _ {j} \ \ \textrm{ and } \ t ^ {m} e ^ {\alpha _ {j} t } \sin \ \beta _ {j} t ,\ m = 0 \dots l _ {j} - 1 . $$

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

$$ f( t) = e ^ {at} ( p _ {m} ( t) \cos bt + q _ {m} ( t) \sin bt ), $$

where $ p _ {m} $ and $ q _ {m} $ are polynomials of degree $ \leq m $, and if the number $ a + b i $ is not a root of (3), one looks for a particular solution of (1) in the form

$$ \tag{5 } x _ {0} ( t) = e ^ {at} ( P _ {m} ( t) \cos b t + Q _ {m} ( t) \sin b t ) . $$

Here $ P _ {m} $ and $ Q _ {m} $ are polynomials of degree $ m $ with undetermined coefficients, which are found by substituting (5) into (1). If $ a + b i $ is a root of (3) of multiplicity $ k $, then one looks for a particular solution of (1) in the form

$$ x _ {0} = t ^ {k} e ^ {at} ( P _ {m} ( t) \cos b t + Q _ {m} ( t) \sin b t ) $$

by the method of undetermined coefficients. If $ x _ {0} ( t) $ is a particular solution of the inhomogeneous equation (1) and $ x _ {1} ( t) \dots x _ {n} ( t) $ is a fundamental system of solutions of the corresponding homogeneous equation (2), then the general solution of (1) is given by the formula

$$ x( t) = x _ {0} ( t) + C _ {1} x _ {1} ( t) + \dots + C _ {n} x _ {n} ( t) , $$

where $ C _ {1} \dots C _ {n} $ are arbitrary constants.

A homogeneous system of linear differential equations of order $ n $,

$$ \tag{6 } \dot{x} = A x , $$

where $ x \in \mathbf R ^ {n} $ is the unknown vector and $ A $ is a constant real $ n \times n $ matrix, can be integrated as follows. If $ \lambda $ is a real eigen value of multiplicity $ k $ of the matrix $ A $, then one looks for a solution $ x = ( x _ {1} \dots x _ {n} ) $ corresponding to $ \lambda $ in the form

$$ \tag{7 } x _ {1} = P _ {1} ( t) e ^ {\lambda t } \dots x _ {n} = P _ {n} ( t) e ^ {\lambda t } . $$

Here $ P _ {1} ( t) \dots P _ {n} ( t) $ are polynomials of degree $ k - 1 $ with undetermined coefficients, which are found by substituting (7) into (6); there are exactly $ k $ linearly independent solutions of the form (7). If $ \lambda $ is a complex eigen value of multiplicity $ k $, then the real and imaginary parts of the complex solutions of the form (7) form $ 2 k $ linearly independent real solutions of (6), and a pair of complex conjugate eigen values $ \lambda $ and $ \overline \lambda \; $ of multiplicity $ k $ of the matrix $ A $ generates $ 2 k $ linearly independent real solutions of (6). Taking all eigen values of $ A $, one finds $ 2 n $ 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 $ X ( t) = e ^ {At} $ is the fundamental matrix of the system (7), normalized at the origin, since $ X ( 0) = E $, the unit matrix. Here

$$ e ^ {At} = E + \sum _ { k= } 1 ^ \infty \frac{A ^ {k} t ^ {k} }{k ! } , $$

and this matrix series converges absolutely for any matrix $ A $ and all real $ t $. Every other fundamental matrix of the system (6) has the form $ e ^ {At} C $, where $ C $ is a constant non-singular matrix of order $ n $.

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)
How to Cite This Entry:
Linear ordinary differential equation with constant coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_ordinary_differential_equation_with_constant_coefficients&oldid=14330
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article