View source for Linear ordinary differential equation with constant coefficients

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An ordinary differential equation (cf. [[Differential equation, ordinary|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|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|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|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====
<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>