9) Linear equations of the n-th order. Basic properties.
Let’s consider the differential equation
=f(x)
(1)
which coefficients p1(x),..., pn (x) and the constant term f (x) are continuous on the interval J ⊆ (− ∞,∞). It is required to find a solution of equation (1) with conditions
y(x0)=y0,
y’(x0)=
, ….
(2)
The problem (1), (2) is called Cauchy problem.
Theorem
1. Let
the coefficients pi
(x),i
=
1,n
of
equation (1) and the right side f
(x)
be
continuous on the interval J .Then it has a unique solution y
=
ϕ(x)∈
(J
)
satisfying
to the initial conditions (2). We introduce a special notation for
the left side of equation (1) L(y)
≡
+p
(x)
pn
(x)y
(3)
and we call L as n order linear differential operator (acting in (J )).
Concept of operator generalizes the notion of function. In this situation, the operator L for each function y∈ (J ) assigns a function L(y)∈V (J ) by the formula (3). Property of the operator L( c1y1 + c2 y2 )= c1L( y1) + c2L( y2 ) , (4)
is called a linearity of operator.If in equation (1) f (x) vanishes, we obtain the equation L(y) = 0 , (5) which is called homogeneous. And the equation (1) is called a linear inhomogeneous. Further, we consider the homogeneous equation (5).
Theorem 2. Linear combination solutions of equation (5) is a solution.
10) Linear homogeneous equations, Solution properties. the Euler method which is a method of constructing a fundamental system of solutions for the equation is
(1)
Following
Euler we find solutions of equation (1) in the form y
=
,
where λ
is
a constant. We prove that equation (1) has a solution of the form y
=
,
if λ
satisfies
to the equation l(λ
)
=
0
, (2) where l(λ)=
Equation (2) is called characteristic for the differential equation (1).
It is considered separately the different cases.
1) The roots of the characteristic polynomial l(λ ) are real and different.
2) The roots of the characteristic polynomial are different, but among of them there are complex roots.
3)
The roots of the characteristic equation are real, but among of them
there are multiple. To multiple root λ
=λ
*
multiplicity
" k
"
corresponding to " k
"
linearly independent solutions of the form
y
1=
,
y
2=x
,
…
=
4) In the general case can also be multiple complex roots. Note that if the multiplicity of the root α + iβ is e , that multiplicity of the adjoint root as well e . Therefore, such pair of roots corresponding to the following linearly independent solutions
eαx cosβx, xeαx cosβx, …….., xl-1 eαx cosβx
eαx sinβx, xeαx sinβx, …….., xl-1 eαx sinβx
11) Linear inhomogeneous equations
Let
consider the equation
which coefficients
and the constant term
are continuous on the interval
.
Its required to find a solution of the equation (1), satisfying to
the initial conditions.
The
equation (1) is called inhomogeneous linear differential equation.
Theorem.
Let the coefficients
of
eq (1) and the right side
be continuous on the interval J. then it has a unique solution
satisfying
to the initial conditions (2). Introduce a special notation for the
left side of the eq(1)
And
we call L
as n order linear differential operator. Concept of operator
generalizes the notation of function. In this situation, the operator
L for each function
assigns a function
by the formula (3) . property of the operator
is called a linearity of operator. If in equation (1)
vanishes, we obtain the equation
which is called homogeneous.
Theorem.
Sum of the solution (1) and any of equation (5) is a solution of (1)
Theorem. (on the structure of the general solution of (1) ) The general solution of inhomogeneous linear differential equation (1) is composed of the general solutions of its corresponding homogeneous equation (5) and any particular solution of (1)