Linear ordinary differential equation with constant coefficients

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)