Lecture no. 27. Systems of differential equations. Approximate integration of differential equations
POINT 1. NORMAL SYSTEMS OF DIFFERENTIAL EQUATIONS
POINT 2. APPROXIMATE INTEGRATION OF DIFFERENTIAL EQUATIONS
Point 1. Normal systems of differential equations
Def. 1. Normal system of differential equations
with n unknown
functions
is called the next system
( 1 )
Def. 2. Solution of
a normal system (1) is called an ordered set of n
functions
which satisfies every equation of the system.
Def. 3. Cauchy problem for a normal system (1) is called a problem of finding solutions of the system which satisfy the next initial conditions
( 2 )
Theorem 1. Let
the functions
and all their first order partial derivatives with respect to
be continuous in some domain D of
the
-dimensional
space
.
Then for any point
Cauchy problem (1), (2) has unique solution.
Def. 4. The
general solution of
a normal system (1) (in the domain D of
the theorem 1) is
called an ordered set of n functions
,
containing n arbitrary
constants, which satisfies two conditions: a) this set is a solution
of the system for any values of
;
b) for any initial conditions (2) (if
)
it is possible to find values
of arbitrary constants to satisfy these conditions.
It can be proved that any system of differential equations, in particular every nth-order differential equation can be reduced to a normal system.
Let, for example, be given the third order differential equation
.
Putting
we get a normal system of equations with three unknown functions
A normal system (1) can often be reduced to one nth-order differential equation with the help of so called elimination method.
Let’s confine ourselves by a normal system of
two linear equations with unknown functions
and constant coefficients
( 3 )
We differentiate the first equation and then substitute the first derivatives of by right sides of the equations of the system,
,
( 4 )
where
.
From
the first equation of the system we find
(if this is possible) and substitute in the equation (4),
,
( 5 )
,
,
( 6 )
where
.
Therefore the system of equations (3) is reduced
to the second order
is its general solution. Finding from (5),
,
we get the general solution of the system (3)
.
Ex. 1. Solve Cauchy problem for a system
with
initial conditions
.
Using the theory we have
.
The general solution of the system
.
Taking into account the initial conditions we get
Answer. The solution of Cauchy problem
.
Ex. 2. Find the general solution of the system
By the theory
,
a)
;
b)
,
.
The general solution of the system
.