36) Solution continuity of parameters and initial data.
Theorem.
Suppose that ƒ and
are continuous and bounded in a given region
U. Let
be
a solution of
(1)
passing through
,
and
be a solution of (1) passing through (
.
Suppose that ϕ
and
ψ
exist on some integral I.
Then,
for each ε > 0, there exists δ > 0 such that if |t
-
|
< δ and ||
||
< δ, then
||ϕ(t) – ψ( )|| < ε, for t, ∈ I.
Proof.
Since ϕ
is the solution of (1) through the point (
),
we have, for all t
∈
I,
ϕ(t)=
+
(2)
As
ψ is the solution of (1) through the point (
,
),
we have, for all t
∈
I,
ψ(t)=
+
(3)
Since
subtracting (3) from (2) gives
||
ϕ(t) – ψ(t) ||
||
||
+ ||
||+||
||
Using the boundedness assumptions on ƒ and to evaluate the right hand side of the latter inequation, we obtain
||
ϕ(t) – ψ(t) ||
||
||
+M
|
|
+ K
If | | < δ, || || <δ, then we have
||
ϕ(t) – ψ(t) ||
+Mδ
+ K ||
||
(4)
Applying Gronwall’s inequality to (4) gives
||ϕ(t)
– ψ(
)||
||ϕ(t)
– ψ(t)||
||ψ(t)
– ψ(
)||
Now,
given ε > 0, we need only choose
ε/[M
+
(1+
]
to obtain the desired inequality, completing the proof.
37. Fundamental system of solutions
Definition.
System of n
linearly
independent solutions
(t),
(t),
…,
(t)
of
system
is called a fundamental system of solutions or basis.
Theorem.
The system
has a fundamental system of solutions. If
(t), (t), …, (t) is basis, then the general solution has the form
where
are
arbitrary constants.
Concept of the fundamental matrix. Ostrogradsky-Liouville formula.
We consider the system
of n arbitrary vector solutions of the vector equation and form a matrix of the order nxn
Y(t)
=
and the Wronskian. If the system of vectors is linearly independent, then detY(t)=W(t)does not vanish for any value t of the interval of continuity of the matrix A(t). In this case the matrix Y(t) is called an integral or a fundamental matrix for the system . If Y( )= E, where E is the unit
40 Integration of linear inhomogeneous equation with quasipolinomial right side.
The particular solution y * (x) can be solved by trial, if the right side of the equation - quasipolynomial - function of the form
f(x) = exp(αx)(Mm(x)cos(βx) + Nn(x)sin(βx)). Here Mm (x) - a polynomial of degree m, Nn (x) - a polynomial of degree n, α and β - are real numbers.
Calculation method of selection of a particular solution of the inhomogeneous linear equations with quasipolynomial in the right side is the following. Carefully look at the right side of the equation and write down the number of α ± βi.
Then the characteristic equation of the homogeneous equation and find its roots. There are two cases: the roots of the characteristic polynomial is no root, equal to the number of α ± βi (nonresonantce case), and among the roots of the characteristic polynomial is r roots equal to the number of α ± βi (resonanCE case).
Consider the non-resonant case (among the roots of the characteristic polynomial is no root, equal to the number of α ± βi). Then the particular solution will search in the form
y*(x) = exp(αx)(Pk(x)cos(βx) + Qk(x)sin(βx)),
We consider the resonance case (among the roots of the characteristic polynomial is r roots equal to the number of α ± βi). Then the particular solution will be sought in the form
y*(x) = exp(αx)(Pk(x)cos(βx) + Qk(x)sin(βx))xr,