system of ordinary differential equations
In many problems one would have to find the functions , , , satisfying the differential equation system of the form
(1) |
Such a system is called normal system. The functions are supposed to be differentiable. Usually there are also given some initial conditions
(2) |
The solving procedure may be as follows.
First we differentiate the first equation (1) with respect to the argument :
Here one substitutes the derivatives as they are given by the equations (1), getting the equation of the form
When one differentiates this equation and makes the substitutions as above, the result has the form
Then one can continue similarly and will finally come to the system
(3) |
The first equations (3) determine as functions of , , :
(4) |
These expressions of are put into the last of the equations (3), and then one has an ’th order differential equation for solving :
(5) |
Solving this gives the function
(6) |
Differentiating this times yields the derivatives as functions of . These derivatives are put into the equations (4), giving the functions :
(7) |
In the solution (6) and (7), one has still to consider the initial
conditions (2); then the constants of integration attain
certain values.
Remark. If the system (1) is linear, then also the
equation (5) is linear.
Example. Solve the functions and from the pair of differential equations
(8) |
subject to the initial conditions and
Differentiation of the first equation with respect to gives
Setting to this the first derivatives from (8) turns it into
(9) |
Into this we put the expression
(10) |
got from the first equation (8), obtaing the second order linear differential equation
with constant coefficients. The general solution of this last equation is
(11) |
and by (10) this yields
(12) |
The initial conditions give from (11) and (12)
whence and and thus the particular solution in question is
References
- 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Teine köide. Viies trükk. Kirjastus Valgus, Tallinn (1966).
Title | system of ordinary differential equations |
---|---|
Canonical name | SystemOfOrdinaryDifferentialEquations |
Date of creation | 2014-03-07 16:37:51 |
Last modified on | 2014-03-07 16:37:51 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 8 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 34A05 |