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Chapter 4. Ordinary differential equations
X
181
__________________________________________________________________________
0.5
0.4
0.3
0.2
0.1
0
-0.1
Figure 4.2.2. Phase portrait for the real part of vector field (4.2.57)
0.05 0.1 0.15
4.2.6. Partial solution for inhomogeneous equation
with harmonic loading
Proposition 4.2.9 (Constructing partial solution)
Let differential equation be written in a form of (4.2.31) with the harmonic loading vector:
0
⎛⎞
⎜⎟
G
...
⎜⎟
where ω is loading frequency, assumed to be real and positive. Then, the partial solution for
Eq. (4.2.31) can be represented in a form
=ω
Pit
⎜⎟
0
⎜⎟
p
⎝⎠
GG
iP
=ω− ⋅IG
()
p
exp
()
−
, (4.2.58)
1
, (4.2.59)
Proof
provided
where
Sp G denotes spectrum of matrix
()
Spiω∉ G
Firstly, it can be noted that at condition (4.2.60) matrix
, (4.2.60)
()
(the complete set of its eigenvalues).
G
iω−IG
()
is invertible, so expression in
the right-hand side of (4.2.59) is well defined.
Now, substituting the assumed solution (4.2.59) into equation (4.2.31) reveals that this equation
is satisfied.
Chapter 4. Ordinary differential equations
X
X
G
X
X
G
p
X
X
G
182
__________________________________________________________________________
Remark 4.2.9
Direct verification shows that the constructed partial solution (4.2.59) generally does not satisfy
initial conditions. More precisely, solution (4.2.59) at
GG
() () exp
ti Pti it
=ω− ⋅ =ω− ⋅ ω
IG IG
p
() ()
00 0
11
−−
Proposition 4.2.10 (Constructing harmonic solution that satisfies initial conditions and inhomogeneous
equation (4.2.31))
G
Let
be a vector, specifying initial conditions at
0
tt=
suppose also that condition (4.2.60) is satisfied, then the solution of Eq. (4.2.31) at harmonic
loading that satisfies initial conditions can be represented in a form:
GG G G
() ( )
te X i Pt i P
()
−
tt
G
= ⋅−ω−⋅ +ω−⋅
0
00
(
IG IG
()
G
X
C
11
−−
satisfies a very peculiar condition
tt=
0
0
⎛⎞
⎜⎟
...
⎜⎟
⎜⎟
0
⎜⎟
p
⎝⎠
, and
0
()
)
()
be harmonic loading (4.2.58),
P
. (4.2.61)
. (4.2.62)
G
X
p
Proof
The first term in the right-hand side of (4.2.62) marked as
solution with initial value
The second term marked as
solution satisfies initial condition (4.2.61). The sum of these terms satisfies the desired
condition
Remark 4.2.10
Very often, when solutions to harmonic loadings are needed, only partial solution (4.2.59) is
considered, leaving initial conditions without satisfying.
Example 4.2.4
Herein we consider the same second-order differential equation, as in Examples 4.2.1 − 4.2.3,
but with harmonic loading vector:
is actually the Cauchy problem
C
GG
−ω− ⋅IG
()
00
G
, is the partial solution for harmonic loading (4.2.58). This
1
−
. (4.2.63)
()Xi Pt
G
0
at
tt=
.
0
0
⎛⎞
it
ω
Pt e
=
()
⎜⎟
5
⎝⎠
, (4.2.64)
with circular frequency
(rad/sec).
7ω=
Chapter 4. Ordinary differential equations
p
G
G
G
183
__________________________________________________________________________
At first stage the partial solution will be constructed. This needs in determining eigenvalues of
matrix G. But, the eigenvalues have actually been obtained, when the Jordan normal form of
was computed in (4.2.42) and (4.2.43):
G
331 331
λ= λ =
12
ii−+ −−
;
44
. (4.2.65)
Comparing (4.2.66) with the loading frequency (multiplied by
iω−IG
()
is invertible. The inversion procedure yields
191 317
ii
−−+
() ()
⎛⎞
1
−
i
ω− =
IG
()
⎜⎟
() ()
⎜⎟
⎝⎠
606 1515
31 7 7 7 31
ii
+−
606 1515
i ) reveals that at 7ω= matrix
. (4.2.67)
Now, according to (4.2.59) multiplication of matrix (4.2.67) by the loading vector yields the
desired partial solution:
191 317 317
ii i
−−+ −+
G
⎜⎟⎜⎟
Xee
=⋅=
p
() ()
⎜⎟⎜⎟
⎝⎠⎝⎠
606 1515 303
317 7731 7731
ii i
+− −
606 1515 303
() ()
⎛⎞⎛⎞
On Fig. 4.2.3 the phase portrait of
Re( )
X
0
⎛⎞
it it
77
⎜⎟
5
⎝⎠
is plotted. It is based on the harmonic loading
()
()
, (4.2.68)
solution (4.2.68) and a parametric plot.
0.6
0.4
0.2
-0.1 -0.08 -0.06-0.04-0.02 0.02 0.04 0.06 0.08 0.1
0
-0.2
-0.4
-0.6
Figure 4.2.3. Phase portrait for the real part of vector field (4.2.68)
Note, that phase portraits related to (partial) solutions of linear differential equations with
constant coefficients and harmonic loadings are always ellipses.
The second stage will be related to constructing the overall solution that should satisfy
homogeneous initial conditions at
:
0t =
0
. (4.2.69)
0X =
0
For the considered case, the first term in the right-hand side of (4.2.62) becomes
Chapter 4. Ordinary differential equations
X
X
G
X
p
X
184
__________________________________________________________________________
⎛⎞
⎜⎟
0
GGG
=⋅−ω−⋅
() ( )
te X i Pt
C
Now, summarizing partial solution (4.2.68) and a homogeneous solution (4.2.70) actually
solving the initial value problem, yields the overall solution that satisfies condition (4.2.69) and
the (inhomogeneous) differential equation (4.2.31):
()
The phase portrait for the real part of the overall solution (4.2.71) is plotted on Fig. 4.2.4.
P
⎜⎟
tt
G
()
−
0
00
⎜⎟
N
G
⎜⎟
0
⎜⎟
⎝⎠
GG
tX X=+
-0.1 -0.05 0.05 0.1 0.15
IG
()
Cp
0.6
0.4
0.2
-0.2
-0.4
-0.6
-0.8
1
−
. (4.2.71)
N
⎛⎞
⎜⎟
⎝⎠
. (4.2.70)
0
5
Figure 4.2.4. Phase portrait for the real part of the overall vector field (4.2.68)
G
Comparing plots on Figs. 4.2.3 and 4.2.4 demonstrates that addition of
G
harmonic solution
noticeably complicates behavior of the resultant solution. However, if
to the partial
C
merely the long-term behavior is interested, it is possible to confine analysis by considering
behavior of partial harmonic solution only.
4.3. Equation Chapter 4 Section 3 Closed form
solutions for linear differential equations with
variable coefficients
Herein several classes of linear differential equations with variable coefficients
admitting closed form solutions, will be considered.
Chapter 4. Ordinary differential equations
x
xxx
x
p
p
p
185
__________________________________________________________________________
4.3.1. Preliminary results
Definition 4.3.1 (Generalized Hamiltonian formalism for equation with variable coefficients)
Let the differential equation be represented in a form
() ( 1)
nn
() () ... () () ()
atx a tx atx atx pt
nn
++++=, (4.3.1)
110
−
−
′
where ( )
() 0
then denoting
at is continuous and
n
x≡ and introducing 1n − new variables
0
at≠ . (4.3.2)
n
=
10
x
=
21
(4.3.3)
.....
x
=
nn
−−
12
the considered equation can be rewritten in terms of these new variables
() ()
at at
10
n
xxx
=− − − +
nn
−−
110
−
() () ()
at at at
nnn
...
()
t
. (4.3.4)
Regrouping terms and combining Eqs. (4.3.3) and (4.3.4) yields the desired system of the first
order equations
xx
=
01
xx
=
12
() ()
at a t
xx x
10 1
nn
−−
01
=− − − +
() () ()
at at at
nnn
.....
xx
=
nn
−−
21
n
...
−
, (4.3.5)
()
t
or in a matrix form
01
xx
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
... ...
d
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
xx
dt
nn
⎜⎟ ⎜⎟
xx
nn
⎝⎠ ⎝⎠
⎛⎞
00
⎜⎟
⎜⎟
⎜⎟
=⋅+
⎜⎟
−−
22
11
−−
at a t
() ()
01
⎜⎟
−− ⎜⎟
⎜⎟
at at
() ()
nn
⎝⎠
...
... ...
1
n
−
⎛⎞
⎜⎟
0
⎜⎟
...
⎜⎟
⎜⎟
⎜⎟
⎜⎟
at
⎝⎠
. (4.3.6)
0
t
()
()
n
Denoting
Chapter 4. Ordinary differential equations
p
X
G
186
__________________________________________________________________________
x
⎛⎞
0
⎜⎟
XP
GG
...
⎜⎟
== =
⎜⎟
⎜⎟
⎝⎠
;;
x
n
−
2
x
1
n
−
⎛⎞
⎜⎟
⎜⎟
⎜⎟
G
⎜⎟
⎜⎟
−− ⎜⎟
⎜⎟
⎝⎠
System (4.3.6) can be rewritten as
01
...
1
() ()
at a t
01
at at
... ...
() ()
nn
n
−
⎛⎞
⎜⎟
0
⎜⎟
...
⎜⎟
⎜⎟
⎜⎟
⎜⎟
at
⎝⎠
. (4.3.7)
0
t
()
()
n
Such a procedure of deriving system (4.3.6) or (4.3.8) from the initial Eq. (4.3.1) is similar to
one used for reducing differential equation with constant coefficients to a system of equations
of the first order, and is known as the generalized Hamiltonian formalism for a linear
order differential equation with variable coefficients.
Assumption 4.3.1
Let matrix
and both
( ) / ( ),..., ( ) / ( )
at at a t at
01
t
nnn
It will be assumed that matrices
interval
T
be an integral:
M
and
belong to a closed interval T, where condition (4.3.2) holds and functions
t
0
−
.
d
=⋅+G
XP
dt
t
() ( )
td=ττ
MG
∫
t
0
are integrable.
and
()tM
GG
. (4.3.8)
-th
n
, (4.3.9)
commute at any t and τ belonging to the
()τG
Proposition 4.3.1
In particular, Assumption 4.3.1 holds, if
where
upon
Λ
G
.
t
Proof
In view of (4.3.10) the matrix integral (4.3.9) can be represented in a form
1
() ()tt
GW WΛ
is the Jordan normal form of ( )tG , i.e. only Jordan normal form
()t
=⋅ ττ⋅
() ( )
td
MW WΛ
−
=⋅ ⋅
1
−
G
t
⎛⎞
⎜⎟
G
∫
⎜⎟
t
0
⎝⎠
, (4.3.10)
, (4.3.11)
depends
Λ
G
Chapter 4. Ordinary differential equations
X
G
G
187
__________________________________________________________________________
ensuring that
definition of the matrix exponent; see Sections 3.6 and 3.7. Actually, matrices
()tM commutes with ()τG . But matrices ()tM and
exp ( )tM
()
commute due to
()tM and
exp ( )tM
()
That implies commutativity of matrices
Proposition 4.3.2
If Assumption 4.3.1 holds, then matrices
Proof
Definition of the matrix exponent ensures that matrices
. But, due to Assumption 4.3.1 matrices
t
flows out that matrices
Remarks 4.3.1
At violating Assumption 4.3.1 Propositions 4.3.1 and 4.3.2 become wrong. In this case
A.
numerical methods for solving the corresponding differential equations are advisable.
B.
A trivial, but important for applications example, when Proposition 4.3.1. and hence
Assumption 4.3.1 are satisfied, correspond to the case, when all the coefficients
at k n=
(), 0,...,
k
are transformed into Jordan normal form by the same transformation matrix
()tM and ()τG at any
()tG and
exp ( )tM
()
of Eq. (4.3.1) are proportional. This ensures
and ( )τG commute due to transitivity of commutation.
()tM
exp ( )tM
()
and
()τG
,tTτ∈
commute at any tT∈ .
()tM and
commute at any
.
exp ( )tM
()
commute at any
. It then
,tTτ∈
W
.
Thus, matrix
previous section.
Proposition 4.3.3
If Assumption 4.3.1 holds, then the general solution of the homogeneous equation (4.3.8) has
the form
where M is the integral matrix defined by (4.3.9) and
defined by initial conditions at
()
at
k
==−
()
at
n
becomes constant one, and we arrive at the situation described in the
G
, 0,..., 1
const k n
. (4.3.12)
4.3.2. Initial value (Cauchy) problem
G
() exp ()
ttC=⋅M
.
t
0
()
, (4.3.13)
is the unknown n-dimensional vector
C
Chapter 4. Ordinary differential equations
X
G
G
X
G
X
X
G
G
G
188
__________________________________________________________________________
Proof
At first we consider the left-hand side of Eq. (4.3.8). Taking derivative of the vector field ( )XtG
in view of (4.3.9) yields
G
dd
() () exp () () exp ()
tt tCttC
=⋅⋅=⋅⋅
dt dt
MMGM
() () ()
t
()
=ττ
d
G
∫
t
0
In obtaining (4.3.14) the assumed commutativity of matrices ( )τG and ( )tM was used.
Now, the right-hand side of the homogeneous Eq. (4.3.8) is
() exp ()ttC⋅⋅GM
()
Comparing (4.3.14) with (4.3.15) completes the proof.
Proposition 4.3.4 (Solution of the initial-value problem)
Let initial conditions at
be defined by vector
tt=
0
the solution of the initial value problem can be represented in a following form
GG
() exp( ())
ttX=⋅M
C
Proof
In view of Proposition 4.3.3, the right-hand side of (4.3.16) is a solution of the homogeneous
Eq. (4.3.8). Now, it remains to note that at
tt=
G
matrix, ensuring
()
tX=
00
C
.
G
. (4.3.14)
. (4.3.15)
, then at fulfillment of Assumption 4.3.1,
0
. (4.3.16)
0
due to (4.3.9) matrix
0
()tM becomes the zero
4.3.3. Partial solution
Proposition 4.3.5 (An auxiliary result for constructing partial solution)
G
Let
be a loading vector, then at Assumption 4.3.1, the equation
P
G
(4.3.17)
GG
Proof
d
dt
() ()tt
−−
MM
eXeP
()
⋅= ⋅
is equivalent to Eq. (4.3.8).
Performing differentiation in the left-hand-side of (4.3.17), yields
() () ()tt t
−− −
−⋅ ⋅ + ⋅ = ⋅
MM M
G
eXe XeP
d
dt
. (4.3.18)
Chapter 4. Ordinary differential equations
X
G
G
X
G
X
X
G
X
G
G
X
189
__________________________________________________________________________
()t
M
e
Now, multiplying both sides of (4.3.18) by a non-singular matrix
()t
commutativity of matrices G and
e
−M
Proposition 4.3.6 (Constructing partial solution)
, we arrive at the initial equation (4.3.8).
and taking into account
Let PG be a loading vector assumed to be a (locally) integrable vector-function of parameter
t ,
then partial solution of Eq. (4.3.8) can be represented in a form
() () ()
te e Pd e Pd
= ⋅ ⋅ττ= ⋅ττ
p
GGG
tt
tt
() () () ( )
MM MM
−τ −τ
∫∫
tt
00
. (4.3.19)
Proof
()t
Integrating both sides of (4.3.17) and multiplying them by a non-singular matrix
M
e
, we
arrive at (4.3.19).
Remark 4.3.2
The constructed partial solution (4.3.19) generally does not satisfy initial conditions. More
precisely, solution (4.3.19) satisfies only homogeneous initial conditions. Actually, it flows out
from (4.3.19) that
() 0
Xt=
0
p
. (4.3.20)
Proposition 4.3.7 (Constructing solution that satisfies initial conditions and inhomogeneous equation
(4.3.8))
G
Let
loading vector, then the solution of Eq. (4.3.8) that satisfies initial conditions can be represented
as the sum of two solutions:
be a vector, specifying initial conditions at
0
G
given by (4.3.16) and
()
t
C
tt=
, and
0
()
p
t
be a (locally) integrable
P
given by (4.3.19). Thus,
Proof
The proof is obvious due to Remark 4.3.2.
Corollary
If initial conditions are homogeneous (
GG G G G
() () () ( )
tXtXte X e Pd
=+=⋅+ ⋅ττ
Cp
GG G
() () ()
tXt e Pd
== ⋅ττ
p
() () ( )
MMM
tt
0
t
MM
∫
t
0
t
0
∫
t
0
), then
0X =
t
() ( )
−τ
−τ
. (4.3.22)
. (4.3.21)
Chapter 4. Ordinary differential equations
[
X
G
X
190
__________________________________________________________________________
Proof
The proof flows out from representation (4.3.21) and Remark 4.3.2.
4.3.4. Lyapunov transformation
Definition 4.3.2 (Lyapunov matrix)
Differentiable in the interval
is called Lyapunov matrix. Matrix norm in (4.3.23) and (4.3.24) can be an arbitrary consistent
matrix norm.
Remark 4.3.3
It can be shown that conditions (4.3.23) and (4.3.24) ensure the inverse matrix to satisfy these
conditions, too.
Definition 4.3.3 (Lyapunov transformation)
A non-degenerate transformation in
;t ∞
)
0
⎛⎞
sup ( ) ( )
⎜⎟
⎝⎠
;
tt
∈∞
)
[
0
sup ( )
tt
∈∞
[
0
matrix
tt
+<∞
LL, (4.3.23)
−
(
;
)
n
\
() () ()
ttYt=⋅L
satisfying conditions
()tL
d
dt
1
t
<∞L (4.3.24)
)
G
(4.3.25)
with Lyapunov matrix
Proposition 4.3.8
Suppose that vector field
transformation (4.3.25), then Eq. (4.3.8) admits the following form in terms of the vector field
G
:
()Yt
Proof
The proof is straightforward and flows out from substituting transformation (4.3.25) into Eq.
(4.3.8).
is called Lyapunov transformation.
()tL
G
in Eq. (4.3.8) relates to the vector field
()
t
GGG
d
YYP
dt
11 1
−− −
=⋅⋅−⋅⋅+⋅LGLLL L
()
. (4.3.26)
G
by Lyapunov
()Yt
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