Variation of constants

A method for solving inhomogeneous (non-homogeneous) linear ordinary differential systems (or equations). For an inhomogeneous system, this method makes it possible to write down in closed form the general solution, if the general solution of the corresponding homogeneous system is known. The idea of the method of variation of constants is that the arbitrary constants participating in the general solution of the homogeneous system are replaced by functions of an independent variable. These functions must be chosen such that the inhomogeneous system is fulfilled. In concrete problems, this method was already applied by L. Euler and D. Bernoulli, but its complete elaboration was given by J.L. Lagrange [1].

Suppose one considers the Cauchy problem for the inhomogeneous linear system

$$ \tag{1 } \dot{x} = A ( t) x + f ( t) ,\ x ( t _ {0} ) = x _ {0} , $$

where

$$ A : ( \alpha , \beta ) \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ), $$

$$ f : ( \alpha , \beta ) \rightarrow \mathbf R ^ {n} $$

are mappings that are summable on every finite interval, and where $ t _ {0} \in ( \alpha , \beta ) $. If $ \Phi ( t) $ is the fundamental matrix solution (cf. Fundamental solution) of the homogeneous system

$$ \tag{2 } \dot{y} = A ( t) y , $$

then $ y = \Phi ( t) c $, $ c \in \mathbf R ^ {n} $, is the general solution of (2). The method of variation of constants consists of a change of variable in (1):

$$ x = \Phi ( t) u , $$

and leads to the Cauchy formula for the solution of (1):

$$ x = \Phi ( t) \Phi ^ {-} 1 ( t _ {0} ) x _ {0} + \Phi ( t) \int\limits _ {t _ {0} } ^ { t } \Phi ^ {-} 1 ( \tau ) f ( \tau ) d \tau . $$

This formula is sometimes called the formula of variation of constants (cf. also Linear ordinary differential equation).

The idea of the method of variation of constants can sometimes be used in a more general non-linear situation for the description of the relation between the solution of a perturbed complete system and that of an unperturbed truncated system (cf. [3], [4]). E.g., for the solution $ x ( t) $ of the problem

$$ \dot{x} = A ( t) x + f ( t , x ) ,\ x ( t _ {0} ) = x _ {0} $$

(where $ A , f $ are continuous mappings and in the case of uniqueness of a solution) the formula of variation of constants is valid. It takes the form of the integral equation

$$ x ( t) = \Phi ( t) \Phi ^ {-} 1 ( t _ {0} ) x _ {0} + \Phi ( t) \int\limits _ {t _ {0} } ^ { t } \Phi ^ {-} 1 ( \tau ) f ( \tau , x ( \tau ) ) d \tau . $$

In it, $ \Phi ( t) $ is the fundamental matrix solution of (2).

References

[1]	J.L. Lagrange, , Oeuvres , 4 , Paris (1869) pp. 151–251
[2]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[3]	V.M. Alekseev, "An estimate for the perturbations of the solutions of ordinary differential equations" Vestnik Moskov. Univ. , 2 (1961) pp. 28–36 (In Russian) (English abstract)
[4]	L.E. Reizin', "Local equivalence of differential equations" , Riga (1971) (In Russian)

References

[a1]	E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1964)
[a2]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a3]	J.K. Hale, "Ordinary differential equations" , Wiley (1980)
[a4]	W.A. Coppel, "Disconjugacy" , Springer (1971)