Mathematics/1.
Differential and integral equations
Dospuloba U.K.
Kostanay state university named after Akhmet Baitursynov, Kazakhstan
Consistency
of solution of differential equations
For consideration
we are given a differential equation of order
where,
– original function of whole-number argument;
– are real constants.
To
find non-trivial (non-zero) solution let’s consider a characteristic equation
Let
be the roots of the characteristic equation.
Here
we can see the following set of solutions:
1)
– real and differential. General solution of the
equation where
are constants of integration, which can be solved
if the initial conditions look like
2) The roots homogeneous equation are real, but among them there are
several multiple roots. For example,
, i.e.
is
-degree multiple root of
homogeneous equation, but all the other
-roots are distinct
ones. General solution of the differential equation is as follows
3) Among the roots of an homogeneous equation there are simple complex
roots. For example,
The
rest roots are real and distinct ones. General solution of differential
equation can be as follows
4) In case of
is
-degree multiple root of a
characteristic equation, then
will be
-degree multiple root and the general
solution of differential will be as follows
Let’s consider inhomogeneous (or non-homogeneous) linear differential equation
of order
are constant real coefficients. General
solution of this equation represents the sum of general solution corresponding
homogeneous equation and any of private solutions of inhomogeneous (or
non-homogeneous) equation.
1)
Let the right part of
inhomogeneous (or non-homogeneous) linear differential equation of order
be as follows
where
– is a multinomial (or polynomial) of
-degree
;
is a real number.
If
is not a root of homogeneous equation the
private solution
is taken from
, where
is multinomial (or polynomial) of
-degree; if
is
-degree root of homogeneous
equation, then
is a multinomial (or polynomial)
-degree.
2)
If the right
part of the equation is given by
or
,
then
the private solution of inhomogeneous (or non-homogeneous) differential
equation looks like
3)
If
or
, then the private solution
of the inhomogeneous (or non-homogeneous) differential equation can be as
follows
The
solution of
differential equation of order
satisfying the initial
conditions
is
called consistent, if for any
it is of the kind when for any other solution of
differential
equation of order
satisfying the entry conditions
of sum of inequations
we get the following inequation
in
any
.
If
in condition of any small
the equation
is not fulfilled for any of
solutions, then
is called inconsistent. If except solving inequation
the condition
is fulfilled, then solution
is called asymptoticly consistent.
Literature:
1 Äåìèäîâè÷ Á. Ï. Ëåêöèè
ïî ìàòåìàòè÷åñêîé òåîðèè óñòîé÷èâîñòè. – Ì.: Íàóêà, 1967, - 472 ñ.