differential equations, fall 2019

Differential EquationsLecture Set 08

Systems of Linear First-Order Differential Equations

林惠勇

Huei-Yung Lin

[email protected]

Robot Vision LabDepartment of Electrical Engineering

National Chung Cheng UniversityChiayi 621, Taiwan

Huei-Yung Lin (RVL, CCUEE) Differential Equations 4152018 1 / 56

First-Order System

A system of n first-order differential equations

dx1

dt= g1(t, x1, x2, . . . , xn)

dx2

dt= g2(t, x1, x2, . . . , xn)

...dxn

dt= gn(t, x1, x2, . . . , xn)

is called a first-order system.

Huei-Yung Lin (RVL, CCUEE) Differential Equations Section 8.1 2 / 56

Linear Systems (1/2)

When each of the functions g1, g2, . . . , gn in the first-order system is

linear in the dependent variables x1, x2, . . . , xn, we get the normal form

of a first-order system of linear equations:1

dx1

dt= a11x1 + a12x2 + · · ·+ a1nxn + f1(t)

dx2

dt= a21x1 + a22x2 + · · ·+ a2nxn + f2(t)

...dxn

dt= an1x1 + an2x2 + · · ·+ annxn + fn(t)

This system is referred simply as a linear system.

1aij and xi are functions of t.


Linear Systems (2/2)

We assume the coefficients aij as well as the functions fi are

continuous on a common interval I.

When fi(t) = 0, i = 1, 2, . . . , n, the linear system is said to be

homogeneous; otherwise it is nonhomogeneous.


Matrix Form of a Linear system

If X, A(t), and F(t) denote the respective matrices

X =

x1(t)x2(t)

...

xn(t)

, A(t) =

a11(t) a12(t) · · · a1n(t)a21(t) a22(t) · · · a2n(t)

......

an1(t) an2(t) · · · ann(t)

, F(t) =

f1(t)f2(t)

...

fn(t)

then the system of linear 1st-order DE can be written as

X′ = AX + F

If the system is homogeneous, its matrix form is then

X′ = AX


Example 1: Systems Written in Matrix Notation

Rewrite the system of linear 1-st order DEs in matrix notation.

dx

dt= 6x + y + z + t

dy

dt= 8x + 7y − z + 10t

dz

dt= 2x + 9y − z + 6t


Solution Vector (1/2)

Definition (8.1.1: Solution Vector)

A solution vector on an interval I is any column vector

X =

x1(t)x2(t)

...

xn(t)

whose entries are differentiable functions satisfying the system

X′ = AX + F

on the interval.


Solution Vector (2/2)

Remark

In the important case n = 2, the equations x1(t) and x2(t) represent a

curve in the x1x2-plane. It is common practice to call a curve in the

plane a trajectory and call the x1x2-plane the phase plane.


Example 2: Verification of Solutions

Verify that on the interval (−∞,∞)

X1 =

(

1

−1

)

e−2t and X2 =

(

3

5

)

e6t

are solutions of

X′ =

(

1 3

5 3

)

X


Initial-Value Problem

Let t0 denote a point on an interval I and

X(t0) =

x1(t0)x2(t0)

...

xn(t0)

and X0 =

γ1

γ2

...

γn

where γi, i = 1, 2, . . . , n are given constants. Then the problem

Solve:

X′ = A(t)X + F(t)

Subject to:

X(t0) = X0

is an initial-value problem on the interval.


Existence of a Unique Solution

Theorem (8.1.1: Existence of a Unique Solution)

Let the entries of the matrices A(t) and F(t) be functions continuous on

a common interval I that contains the point t0. Then there exists a

unique solution of the initial-value problem on the interval.


Superposition Principle

Theorem (8.1.2: Superposition Principle)

Let X1,X2, . . . ,Xk be a set of solution vectors of the homogeneous

system

X′ = AX

on an interval I. Then the linear combination

X = c1X1 + c2X2 + · · ·+ ckXk

where ci, i = 1, 2, . . . , k are arbitrary constants, is also a solution on the

interval.


Example 3: Using the Superposition Principle

Verify that

X1 =

cos t

− 1

2cos t + 1

2sin t

− cos t − sin t

, X2 =

0

et

0

, X3 = c1X1 + c2X2

are solutions of the system

X′ =

1 0 1

1 1 0

−2 0 −1

X


Linear Dependence/Independence

Definition (8.1.2: Linear Dependence/Independence)

Let X1,X2, . . . ,Xk be a set of solution vectors of the homogeneous

system

X′ = AX

on an interval I. We say that the set is linearly dependent on the

interval if there exist constants c1, c2, . . . , ck, not all zero, such that

c1X1 + c2X2 + · · ·+ ckXk = 0

for every t in the interval. If the set of vectors is not linearly dependent

on the interval, it is said to be linearly independent.


Criterion for Linear Independent Solutions

Theorem (8.1.3: Criterion for Linear Independent Solutions)

Let X1 =

x11

x21

...

xn1

, X2 =

x12

x22

...

xn2

, . . . , Xn =

x1n

x2n

...

xnn

be n

solution vectors of the homogeneous system X′ = AX on an interval I.

Then the set of solution vectors is linearly independent on I if and only

if the Wronskian

W(X1,X2, . . . ,Xn) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

x11 x12 · · · x1n

x21 x22 · · · x2n

......

xn1 xn2 · · · xnn

∣

∣

∣

∣

∣

∣

∣

∣

∣

6= 0

for every t in the interval.


Wronskian

It can be shown that if X1,X2, . . . ,Xn are solution vectors of

X′ = AX

then for every t in I either W(X1,X2, . . . ,Xn) 6= 0 or

W(X1,X2, . . . ,Xn) = 0.

Thus if we can show that W 6= 0 for some t0 in I, then W 6= 0 for every t,

and hence the solutions are linearly independent on the interval.


Example 4: Linearly Independent Solutions

Verify that the solutions

X1 =

(

1

−1

)

e−2t and X2 =

(

3

5

)

e6t

of the system

X′ =

(

1 3

5 3

)

X

given in Slide (9) are linearly independent solutions.


Existence of a Fundamental Set of Solutions

Theorem (8.1.4: Existence of a Fundamental Set)

Any set X1,X2, . . . ,Xn of n linearly independent solution vectors of the

homogeneous system

X′ = AX

on an interval I is said to be a fundamental set of solutions on the

interval.

There exists a fundamental set of solutions for the homogeneous

system

X′ = AX

on an interval I.


General Solution – Homogeneous Systems

Theorem (8.1.5: General Solution – Homogeneous Systems)

Let X1,X2, . . . ,Xn be a fundamental set of solutions of the

homogeneous system

X′ = AX

on an interval I. Then the general solution of the system on the

interval is

X = c1X1 + c2X2 + · · ·+ cnXn

where ci, i = 1, 2, . . . , n are arbitrary constants.


Example 5: General Solution

Find the general solution of

X′ =

(

1 3

5 3

)

X


Example 6: General Solution


X′ =

1 0 1

1 1 0

−2 0 −1

X


General Solution – Nonhomogeneous Systems

Theorem (8.1.6: General Solution – Nonhomogeneous Systems)

Let Xp be a given solution of the nonhomogeneous system

X′ = AX + F (1)

on an interval I, and let Xc = c1X1 + c2X2 + · · ·+ cnXn denote the

general solution on the same interval of the associated homogeneous

system

X′ = AX (2)

Then the general solution of the nonhomogeneous system on the

interval is X = Xc + Xp.

The general solution Xc of the associated homogeneous system (2) is

called the complementary function of the nonhomogeneous system

(1).


Example 7: General Solution – Nonhomogeneous

System


X′ =

(

1 3

5 3

)

X +

(

12t − 11

−3

)


Eigenvalues and Eigenvectors (1/2)

Let

X = (k1 k2 · · · kn)⊤

eλt = Keλt

be a solution vector of the general homogeneous linear first-order

system

X′ = AX (3)

then

λKeλt = AKeλt

That is,

(A− λI)K = 0 (4)

Note: The symbol ⊤ is a transpose, and (k1 k2 · · · kn)⊤ is a column

vector in a more compact form.


Eigenvalues and Eigenvectors (2/2)

If the system has a nontrivial solution, then we must have

det(A− λI) = 0

The polynomial equation in λ is called the characteristic equation of

the matrix A; its solutions are the eigenvalues of A.

A solution K 6= 0 of Eq. (4) corresponding to an eigenvalue λ is called

an eigenvector of A.

A solution of the homogeneous system2

X′ = AX

is then X = Keλt.

2Plug in the solution to the homogeneous system to verify!


General Solution – Homogeneous Systems

Theorem (8.2.1: General Solution – Homogeneous Systems)

Let λ1, λ2, . . . , λn be n distinct real eigenvalues of the coefficient matrix

A of the homogeneous system

X′ = AX

and let K1,K2, . . . ,Kn be the corresponding eigenvectors.

Then the general solution of

X′ = AX

on the interval (−∞,∞) is given by

X = c1K1eλ1t + c2K2eλ2t + · · ·+ cnKneλnt


Example 1: Distinct Eigenvalues

Solve

dx

dt= 2x + 3y

dy

dt= 2x + y


Example 2: Distinct Eigenvalues

Solve

dx

dt= −4x + y + z

dy

dt= x + 5y − z

dz

dt= y − 3z


Repeated Eigenvalues (1/2)

Not all of the n eigenvalues λ1, λ2, . . . , λn for an n × n matrix A need be

distinct.

In general, if m is a positive integer and (λ− λ1)m is a factor of the

characteristic equation while (λ− λ1)m+1 is not a factor, then λ1 is said

to be an eigenvalue of multiplicity m.

For some n × n matrices A it may be possible to find m linearly

independent eigenvectors K1,K2, . . . ,Km corresponding to an

eigenvalue λ1 of multiplicity m ≤ n. In this case the general

solution of the system contains the linear combination

c1K1eλ1t + c2K2eλ1t + · · ·+ cmKmeλ1t


Repeated Eigenvalues (2/2)

If there is only one eigenvector corresponding to the eigenvalue λ1

of multiplicity m, then m linearly independent solution of the form

X1 = K11eλ1t

X2 = K21teλ1t + K22eλ1t

· · · · · ·

· · · · · ·

Xm = Km1

tm−1

(m − 1)!eλ1t + Km2

tm−2

(m − 2)!eλ1t + · · ·+ Kmmeλ1t

where Kij are column vectors, can always be found.


Example 3: Repeated Eigenvalues

Solve

X′ =

1 −2 2

−2 1 −2

2 −2 1

X


Second Solution

Suppose that λ1 is an eigenvalue of multiplicity two and that there is

only one eigenvector associated with this value. A second solution can

be found of the form

X2 = Kteλ1t + Peλ1t (5)

where

K =

k1

k2

...

kn

and P =

p1

p2

...

pn


Example 4: Repeated Eigenvalues

Find the general solution of the system

X′ =

(

3 −18

2 −9

)

X

Note: Do not use Eqs. (13) and (14) in the textbook!!!


Eigenvalue of Multiplicity Three

When the coefficient matrix A has only one eigenvector associated

with an eigenvalue λ1 of multiplicity three, we can find a second

solution of the form (5) and a third solution of the form

X3 = Kt2

2eλ1t + Pteλ1t + Qeλ1t (6)

where

K =

k1

k2

...

kn

, P =

p1

p2

...

pn

, Q =

q1

q2

...

qn


Example

Verify that the general solution of

dx

dt= 6x − y

dy

dt= 5x + 4y

is

X = c1

(

1

1 − 2i

)

e(5+2i)t + c2

(

1

1 + 2i

)

e(5−2i)t


Solutions Corresponding to a Complex Eigenvalue

Theorem (8.2.2: Solutions Corresponding to a ComplexEigenvalue)

Let A be the coefficient matrix having real entries of the homogeneous

system

X′ = AX

and let K1 be an eigenvector corresponding to the complex eigenvalue

λ1 = α+ iβ, where α and β are real.

Then

K1eλ1t and K1eλ1t

are solutions of the system X′ = AX.

Remark

Check the previous example!


Real Solutions Corresp. to a Complex Eigenvalue

Theorem (8.2.3: Real Sol. Corresp. to a Complex Eigenvalue)

Let λ1 = α+ iβ be a complex eigenvalue of the coefficient matrix A in

the homogeneous system X′ = AX, and let B1 and B2 denote the

column vectors defined by

B1 = Re(K1) and B2 = Im(K1)

Then

X1 = eαt[B1 cosβt − B2 sinβt]

X2 = eαt[B2 cosβt + B1 sinβt]

are linearly independent solution of X′ = AX on (−∞,∞).

Note: Recall the Euler’s formula: eiθ = cos θ + i sin θ.

(Check page 350 in the textbook for proof.)


Example 6: Complex Eigenvalues

Solve the initial-value problem

X′ =

(

2 8

−1 −2

)

X, X(0) =

(

2

−1

)


Nonhomogeneous Linear Systems

The methods of undetermined coefficients and variation of

parameters used to find particular solutions of nonhomogeneous

linear ODEs can both be adapted to the solution of

nonhomogeneous linear systems.

Of the two methods, variation of parameters is the more powerful

technique. However, there are instances when the method of

undetermined coefficients provides a quick means of finding a

particular solution.


Undetermined Coefficients

The method of undetermined coefficients consists of making an

educated guess about the form of a particular solution vector Xp; the

guess is motivated by the types of functions that make up the entries

of the column matrix F(t).

The matrix version of undetermined coefficients is applicable to

X′ = AX + F(t)

only when the entries of A are constants and the entries of F(t) are

constants, polynomials, exponential functions, sines and cosines, or

finite sums and products of these functions.


Example 1: Undetermined Coefficients

Solve the system

X′ =

(

−1 2

−1 1

)

X +

(

−8

3

)

on (−∞,∞)


Example 3: Form of Xp

Determine the form of a particular solution vector Xp for the system

dx

dt= 5x + 3y − 2e−t + 1

dy

dt= −x + y + e−t − 5t + 7


A Fundamental Matrix (1/2)

If X1,X2, . . . ,Xn is a fundamental set of solutions of the homogeneous

system

X′ = AX

on an interval I, then its general solution on the interval is the linear

combination X = c1X1 + c2X2 + · · ·+ cnXn or

X = c1

x11

x21

...

xn1

+ c2

x12

x22

...

xn2

+ · · ·+ cn

x1n

x2n

...

xnn

=

c1x11 + c2x12 + · · ·+ cnx1n

c1x21 + c2x22 + · · ·+ cnx2n

...

c1xn1 + c2xn2 + · · ·+ cnxnn


A Fundamental Matrix (2/2)

It can also be written as

X = Φ(t)C

where C is an n × 1 column vector of arbitrary constants, c1, c2, · · · , cn,

and the n × n matrix

Φ(t) =

x11 x12 · · · x1n

x21 x22 · · · x2n

......

xn1 xn2 · · · xnn

is called a fundamental matrix of the system in the interval.


Properties of Fundamental Matrix

There are some important properties of the fundamental matrix:

A fundamental matrix Φ(t) is nonsingular.

The inverse Φ−1(t) exists for every t in the interval.

If Φ(t) is a fundamental matrix of the system X′ = AX, then

Φ′(t) = AΦ(t) (7)


Variation of Parameters (1/2)

Check if it is possible to replace the matrix of constants C in X = Φ(t)Cby a column vector of functions U(t) = (u1(t) u2(t) · · · un(t))

⊤ so that

Xp = Φ(t)U(t) (8)

is a particular solution of the nonhomogeneous system3

X′ = AX + F(t) (9)

From Eq. (8), ⇒ X′p = Φ(t)U′(t) + Φ′(t)U(t)

From Eq. (9), ⇒ Φ(t)U′(t) + Φ′(t)U(t) = AΦ(t)U(t) + F(t)

From Eq. (7), ⇒ Φ(t)U′(t) + AΦ(t)U(t) = AΦ(t)U(t) + F(t)

⇒ Φ(t)U′(t) = F(t) ⇒ U′(t) = Φ−1(t)F(t) ⇒ U(t) =∫

Φ−1(t)F(t)dt

Thus, Xp = Φ(t)U(t) = Φ(t)∫

Φ−1(t)F(t)dt

3Check Chapter 4, Slide 61.


Variation of Parameters (2/2)

A particular solution of X′ = AX + F(t) is

Xp = Φ(t)

∫

Φ−1(t)F(t)dt

The general solution is X = Xc + Xp, or

X = Φ(t)C +Φ(t)

∫

Φ−1(t)F(t)dt


Example 4: Variation of Parameters

Solve the system

X′ =

(

−3 1

2 −4

)

X +

(

3t

e−t

)

on (−∞,∞)


Matrix Exponential (1/2)

Definition (8.4.1: Matrix Exponential)

For any n × n matrix A, the matrix exponential is defined as

eAt = I+ At + A2

t2

2!+ · · ·+ A

k tk

k!+ · · · =

∞∑

k=0

Ak tk

k!


Matrix Exponential (2/2)

Remark

The derivative of the matrix exponential is given bya

d

dteAt = AeAt

Thus,

X = eAtC

is a solution of X′ = AX.

The function Ψ(t) = eAt is a fundamental matrix of the system X′ = AX.

aBy definition 8.4.1.


Nonhomogeneous Systems

For a nonhomogeneous system of linear first-order differential

equations

X′ = AX + F(t)

where A is an n × n matrix of constants, the general solution is given by

X = Xc + Xp = eAtC + eAt

∫ t

t0

e−AτF(τ)dτ


Computation of eAt

The matrix eAt can be computed by the Laplace transform:

eAt = L−1

{

(sI− A)−1}

since it is a solution of the initial-value problem

X′ = AX and X(0) = I (10)

x(s) = L {X(t)} = L{

eAt}

From the IVP Eq. (10) ⇒ sx(s)− X(0) = Ax(s)

⇒ (sI− A)x(s) = I

⇒ x(s) = (sI− A)−1I = (sI− A)−1

⇒ L{

eAt}

= (sI− A)−1 or eAt = L−1

{

(sI− A)−1}


Example 1: Matrix Exponential

Use the Laplace transform to compute eAt for

A =

(

1 −1

2 −2

)


Homework

Exercises 8.1: 4, 13, 22.

Exercises 8.2: 7, 14, 25, 31, 44.

Exercises 8.3: 5, 16, 21, 32.

Exercises 8.4: 4, 11, 22.


Appendix: Review of Eigensystem (1/2)

Definition

The eigenvalues of a p × p real or complex matrix A are the real or

complex numbers λ for which there is a nonzero x 6= 0 with Ax = λx.

The eigenvectors of A are the nonzero vectors x 6= 0 for which there is

a number λ with Ax = λx. If Ax = λx for x 6= 0, then x is an eigenvector

associated with the eigenvalue λ, and vice versa.

The associated eigenvalues and eigenvectors together make up the

eigensystem of A.


Appendix: Review of Eigensystem (2/2)

Theorem

λ is an eigenvalue of A if and only if A− λI is singular, which in turn

holds if and only if the determinant of A− λI equals zero:

det(A− λI) = 0 (the so-called characteristic equation of A).


differential equations, fall 2019

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