Lecture 6

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Lecture 6

Systems of Differential Equations

So far, we have only considered differential equations with one dependent variable and one

independent variable. In other words, they are differential equations whose solutions are in the form of

  x

y

where y is the dependent variable and

x

is the independent variable.

Nevertheless, many scientific and engineering phenomena may involve two or more dependent

variables, while each of them is a function of the same independent variable. For example, populations of prey and predator can be characterized by variables

x

1 and

x

₂ respectively, while both are functions of time

t

. Changing in

x

₁ or

x

₂ will affect the other as the predator cannot live without the prey, but the prey can live without the predator. Thus, if we write one differential equation to describe change in the prey population and another differential equation to describe change in the predator population, we would expect something like

4 1 3 2

2 2 1 1

a x dt a dx

a x dt a

dx











(1a,b)

where

a

₁,

a

₂,

a

₃,

a

₄ are some positive constants. As we can see,

x

₂ negatively affect the change in

x

₁, while

x

₁ positively affect the change in

x

₂. Certainly, the real situation is more complex than this. For instance, there has to be a limit of how much

x

₂ population can grow since at some point there will not be enough prey for them, and they would start to die out. Equations (1a) and (1b) will need to be modified to include such effect.

Equations (1a) and (1b) make up a system of differential equations. Both equations are needed in order to completely explain what happen in the system.

In order to evaluate a physical system, it may sometimes be easier to separate the system into several subsystems, each with its own differential equation. For instance, if we consider transfer of water between two tanks, each tank can be a subsystem with a differential equation describing rate of change of water going in and out of it.

Rate of change of neutron in a reactor can be divided into rate of change of prompt neutrons and rate of change of delayed neutrons.

Water in a reactor sometimes constitute of both liquid and vapor phases. Each of these phases can be described by a differential equation. However, since the two phases can change back and forth, the differential equations cannot be solved separately.

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Analogous to the system of algebraic equations, in order to solve the system of differential equations with

n

dependent variables,

n

differential equations are needed.

First-Order System

First-order system is a system that consists of all first-order differential equations.

Any higher-order system can be converted into first-order system by method of substitution of variables. Consider the system

 



₁ ₂ ₁ ₂



2 2

2 1 2 1 1 1

, , , ,

x x x x t f x



 





 



(2a,b)

which consists of all second-order differential equations, we can let

x

3

 x

1



and

x

4

 x

2



, and rewrite the system as

 



1 2 3 4



2 2 4

4 3 2 1 1 1 3

4 2

3 1

, , , ,

x x x x t f x x

x x

 

 

 

 

(3a,b,c,d)

which now consists of all first-order differential equations.

As we can see, n-order differential equation can be converted into n first-order differential equations.

Example: x2x5xx4 can be rewritten in the form of first-order system as

4 5

2 ₃ ₂ ₁

3 3 2

2 1









 



x x x x

x x



x x x

x x

 

 



2 3

1 2 1

Solving System of Differential Equations

The main reason why one would want to convert higher-order differential equation into first-order system is because the system can be solved effectively using variety of available numerical algorithms such as Newton’s method or Runge-Kutta method.

Since all of the differential equations in the system of differential equations have to be solved

simultaneously, the most effective way is to solve them on a computer. With this being said, however, there are several analytical techniques that we can use to solve some types of systems of differential equation by hand.

Graphical Solution of Two-Dimensional Systems

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For a system with two dependent variables, says

x

and y, one may choose to plot the solution of the system graphically on the

xy

coordinate axes. Since both

x

and

y

are functions of the same

independent variable, says

t

, their values change as

t

changes. The plot then represents trajectory with arrow pointing toward increasing direction of

t

.

One may choose to plot

x   t

and

y   t

separately in order to see how

x

and

y

change.

For instance, consider a system

y x y

y x

2 . 0 01 .

1 







Its solutions are

t Ae

y

t Ae

x

t t

t

cos 10 sin

1

sin

10 / 10

/ 10 /







Suppose A1, then the graphical solutions are [C. H. Edwards Jr., Differential Equations Computing and Modeling, 1996, p. 226]

Linear Systems

Consider a system of linear first-order differential equations

n n nn n

n n

f x a x

a x a x

f x a x

a x a x

f x a x

a x a x



 



 



 







2 2 1 1

2 2

2 22 1 21 2

1 1

2 12 1 11 1

(4)

where a_ij and

f

_i can be either constants or functions of

t

.

If all

f

_i’s are zero, then the system is homogeneous. Otherwise, it is nonhomogeneous.

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To find unique solution for the system of

n

first-order differential equations,

n

initial conditions are needed. i.e.

 

₁

1

a b

x 

,

x

₂

  a  b

₂, …,

x

_n

  a  b

_n (5)

Notice that this is the same as saying to find unique solution for n-order differential equation, n initial conditions are needed.

Method of Elimination

For small first-order systems, e.g. two- and three- dimensional systems, the easiest way to solve the systems is to combine all of the differential equations together into one higher-order differential equation, and use method discussed in the last lecture to solve it.

This is basically the opposite of how we convert the higher-order differential equation into system of first-order differential equations. Multiple dependent variables can be eliminated until only a single variable is left. For instance, consider a system

z y z

y x y

z y x x



 



 



 

2 (6a,b,c)

The goal is to eliminate dependent variables y and z. By differentiating (6a), we get

z y x

x    (7)

Substituting (6b) and (6c) into (7) yields

 x y   y z  x x y z x

x    

2

     

2



2



(8)

Differentiating (8) again yields

z y x x

x    

2

 

2

  

(9)

Substitute (6b) and (6c) into (8) yields

 x y   y z  x x x y z x

x

x    

2

 

22

     

2

 

4



3



(10)

Multiplying (8) by 2 and subtract (6a) from it yields

z y x x z y x z y x x x

x    

2

 

4



4



2

   

2

 

3



3



2 (11)

Subtract (11) from (10) gives

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x x z y x x z y x x x x x

x2   2 4 3  2 3 3    (12) Rearranging gives

0 3   

 x x x

x (13)

Equation (13) only has one dependent variable left which we can readily solve using method described before. Once

x

is found, we can substitute it to (6b) and solve for

y

, then substitute

y

in (6c) and solve for

z

. Alternatively, we can also eliminate

x

and

z

, and find differential equation in term of

y

; or eliminate

x

and y, and find differential equation in term of z.

Linear Differential Operators Let’s define operator

n n n

dt D d dt

D  d  

(14)

and operator

0 1 1

1D aD a

a D a

L _n ⁿ _n_ ⁿ^   , (15)

where a_n,a_n_₁,...,a₀ are constants.

Operator

L

is called linear differential operator since such operation possesses linear properties:



 L

1

 L

2

 x  L

1

x  L

2

x



 L

1

L

2

 x  L

1

  L

2

x



L

₁

  L

₂

x  L

₂

  L

₁

x

Note that it is important that all of the coefficients in (15) are constants for these properties to be true.

If the coefficients are functions of

t

, for instance, the last properties will no longer be true, and the method below will not work.

Consider a system of two linear differential equations. One can write it using linear differential operator as

2 4 3

1 2 1

f y L x L









(16a,b)

To eliminate the

x

term, we multiply (16a) by

L

₃, and subtract it from the product of

L

₁ and (16b).

Thus,

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

L1L4L3L2



yL1f2L3f1 (17) Notice that (17) is equivalent to writing

2 3

1 1 4

3 2 1

f L

f y L

L L

L

L 

(18)

It turns out that if we eliminate y instead of

x

, we would instead get

4 2

2 1 4 3

2 1

L f

L x f

L L

L

L 

(19)

For higher dimensional linear system, it can be shown that if the system is

n n nn n

n

n n

f x L x

L x L

f x L x

L x L

f x L x

L x L



















2 2 1 1

2 2

2 22 1 21

1 1

2 12 1 11

⁽²⁰⁾

then

nn n

n

n n

nn n

n

n n

L L

f

L L

f

L L

f x L L

L

L L

L

L L

L



















2

2 22

2

1 12

1

2 1

2 22

21

1 12

11



(21)

Or Lijxk= determinant of coefficient matrix whose k^th column is replaced by vector



f₁ f₂  fn



^T

Note that this method may be useful for low dimension system. For higher dimension, finding determinant can be troublesome.

Writing Linear System

It may be more convenient to write a linear system using vector and matrix notations. The system in (4) can be written as

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 





 







 





 





 





 







 





 





n n nn n

n

n n

n

f

f f

x x x

a a

a

a a

a

a a

a

x x x

dt d















2 1 2 1

2 1

2 22

21

1 12

11 2

1

(22)

 

Ax f dt

x

d   (23)

Homogeneous Linear System

For a homogeneous system of n dimensions, there exist n linearly independent solutions x1,x2,...,xⁿ, and the general solution can be written as

n

x

n

c x

c x c

x 

1 1



2 2



...



(24)

where c₁,c₂,...,cn are constants.

Example: Solutions of

dt x

x

d 



 







 

7 6

3 4

are





 



 

^t_t

e x e

₂

2

1 2

3 and





 



 

^_^t_t

e

x e

₅

5

2 3 . Thus the general solution can be written as

 

 







 

 

 



 

 

 



 

^_ _t ^_ _t

t t

t t t

t

e c e c

e c e c e

c e e c e

x

₅

2 2 1

5 2 2 1 5

5 2 2

2

1 2 3

3 3

2 3

Consequently, x₁ 3c₁e²^tc₂e^⁵^t and x₂ 2c₁e²^t 3c₂e^⁵^t

If

 





 







ni i i

i

x x x

x



2 1

then, to check for linear independency of (24), we find its corresponding Wronskian

nn n

n

n n

x x

x

x x

x

x x

x W











2 1

2 22

21

1 12

11



(25)

(8)

Then

 If W 0 on I, they are linearly dependent.

 If W 0 at each point on I , then they are linearly independent.

Nonhomogeneous System

For nonhomogeneous system, the solutions consist of complementary solution

x

_c

and particular solution x_p

, and the general solution has the form

p

c x

x

x  (26)

x

c

 is essentially the general solution of the corresponding homogeneous system. Thus, according to (24), the general solution for nonhomogeneous system would have the form

p

n

x

n

x

c x

c x c

x 

1 1



2 2



...

 

(27)

Eigenvalue Method for Homogeneous Systems

This method works if all of the coefficients in the system are constants. Recall that for a linear system with constant coefficient, we can “guess” the solution to be of exponential form. Let the solution be written as

t t

n t n

t t

n

e v e v v v

e v

x x x

x

^ ^





 





 







 







 









 





 







  

2 1 2

1 2 1

(28)

Then,

 

^v^e ^t ^v^e ^t

dt d dt

x

d  _ 



_ (29)

From (23), we can write

   

Ax Ave ^t dt

x

d   _ . (30)

Then,

 

Ave^^t 



ve^^t (31)

By canceling the non-zero exponential factor and rearranging the terms, we arrive at

(9)

   



A 



I



v0 (32)

For the solution to be nontrivial, determinant of the term in the parentheses must be zero.

   

 

⁰

det

A   I 

(33)

As we have seen before, for

n

equations, there will be

n

eigenvalues (



), and each eigenvalue will have its corresponding eigenvector (

v

).

In addition, there will be situations when the eigenvalues become complex or they are repeated. The following examples will show you how to deal with each case.

Example: Distinct real eigenvalues: C.H. Edwards Jr., Differential Equations Computing and Modeling, 1996, New Jersey: Prentice-Hall, Inc., Page 279-280.

Example: Distinct complex eigenvalues: C.H. Edwards Jr., Differential Equations Computing and Modeling, 1996, New Jersey: Prentice-Hall, Inc., Page 284-285.

Example: Repeated eigenvalues: C.H. Edwards Jr., Differential Equations Computing and Modeling, 1996, New Jersey: Prentice-Hall, Inc., Page 305-307.

It is also possible that in some cases, repeated eigenvalues would no yield additional eigenvector. These eigenvalues are called defective eigenvalues.

It turns out that there is a specific way to deal with this situation.

For two-dimensional system, if both eigenvalues have the same eigenvalue



, and one of them is defective (i.e. only one eigenvector can be found), then the linearly independent solutions will have the form

e

t

v

x

1



1 ^ and ^x² ^



^v¹^t^^v²



^e^^t ^(34a,b)

where

v

1 and

v

2 satisfy

   



^A ^



^I



²^v² ^⁰ and

    

^A ^



^I



^v² ^^v¹ (35a,b)

Example: Repeated eigenvalues: C.H. Edwards Jr., Differential Equations Computing and Modeling, 1996, New Jersey: Prentice-Hall, Inc., Page 309.

Vector

v

2 is known as generalized eigenvector. That is, if



is an eigenvalue, then its corresponding rank k generalized vector

v

^k is such that

   



^A ^



^I



^k^v^k ^⁰ but

    

^A ^



^I



^k^¹^v^k ^⁰ (36a,b) And

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   

 

   

 

   

 

0

1 1 2

1











 

v I A

v v I A

v v I

A k k



 (37)

(Note that if k 1, we recover (32) and this is just ordinary eigenvector.) The linearly independent solutions for the k defective chain are

 

^k ^k ^k ^t

k k

t t

t

e v t t v

v k

t x v

e v t v t v x

e v t v x

e v x



 

 



    

 

 

 



  







 



1 2 2 1

1

3 2 2 1 3

2 1 2

1 1

! 2

! 1 2 1





(38)

Example: Repeated eigenvalues: C.H. Edwards Jr., Differential Equations Computing and Modeling, 1996, New Jersey: Prentice-Hall, Inc., Page 311.

Nonlinear System

Although numerical tools can be used to solve nonlinear system, there are some behaviors of the system which we can sometimes tell from looking at the equations.

Phase portrait is a good way to analyze nonlinear system.

Suppose we have a system

 

  x y dt G

dx

y x dt F dy

, ,



We can see that

    x y G

y x F dt dx

dt dy dx dy

, , /

/



.

If we plot this in xy coordinate, we can construct directional field, a.k.a. phase portrait as before.

For instance, a system with

(11)

1

x

2

y

y x x



 



 

has the following phase portrait. [C. H. Edwards Jr., Differential Equations Computing and Modeling, 1996, p. 335]

There are 2 critical points or nodes in the portrait. One at

 

1,1 and the other at

 

1,



1



. Both of these nodes are unstable because there are arrows pointing away from them. Thus, if the system is moved a little bit away from either of the node, it will start to deviate further away from that node.

Point

 

1,1 is called saddle point as some of the arrows point toward it, and some point away from it.

A node is stable if the arrows around it neither point toward nor away from it, e.g. circle around it.

A node is asymptotically stable if all the arrows around it point toward the node.