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2 Systems of Equations: Preliminary Definitions and Theory

2 Systems of Equations: Preliminary Definitions and Theory

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428



Chapter 6 Systems of Ordinary Differential Equations



2

1

5



10



15



20



25



-1

-2



Figure 6-1 In the limit as t



, the solution is periodic



and the initial-value problem is equivalent to the initial-value problem

y



x

y



1



x0



x2 y

1, y 0



x

1.



We use NDSolve to generate a numerical solution to this initial-value problem

valid for 0 t 25.

In[1091]:= numsol

NDSolve



x t



y t



1



Out[1091]=



y t ,

x t



2



y t



x t ,



x 0

1, y 0

1 , x t , y t , t, 0, 25

x t

InterpolatingFunction

0., 25. , <> t ,

y t

InterpolatingFunction

0., 25. , <> t



We can use this result to approximate the solution for various values of t. For

example, entering

In[1092]:=

Out[1092]=



x t ,y t



/. numsol /. t > 1



1.29848, 0.367035



shows us that x 1

1.29848 and x 1

y1

0.367035. We use Plot to graph

x t and y t (the graph of y t is in gray) in Figure 6-1.

In[1093]:= Plot Evaluate x t , y t /. numsol , t, 0, 25 ,

PlotStyle > GrayLevel 0 , GrayLevel 0.5



6.2 Systems of Equations: Preliminary Definitions and Theory



3

2

1



-3



-2



-1



1



2



3



-1

-2

-3

Figure 6-2



The solution approaches a limit cycle



Because we let x

y, notice that y t > 0 when x t is increasing and y t < 0

when x t is decreasing. The observation that these solutions are periodic is further confirmed by a graph of x t (the horizontal axis) versus y t (the vertical axis)

generated with ParametricPlot in Figure 6-2. We see that as t increases, the

solution approaches a certain fixed path, called a limit cycle.

In[1094]:= ParametricPlot x t , y t /.numsol, t, 0, 25 ,

PlotRange >

3, 3 , 3, 3 ,

AspectRatio > 1, Compiled > False



We will find that nonlinear equations are more easily studied when they are written as a system of equations.



6.2.1 Preliminary Theory

Definition 24 (System of Ordinary Differential Equations). A system of ordinary

differential equations is a simultaneous set of equations that involves two or more dependent variables that depend on one independent variable. A solution of the system is a set

of functions that satisfies each equation on some interval I.



429



430



Chapter 6 Systems of Ordinary Differential Equations



If the differential equations in the system of differential equations are linear equations, we say that the system is a linear system of differential equations or a

linear system.



EXAMPLE 6.2.1: Show that

tion to the system



x



y



y



5x



x

y



1 t

5e

t



e



et



cos 2t



cos 2t



3 sin 2t



sin 2t



is a solu-



0

2y



1.



SOLUTION: The set of functions is a solution to the system of equations because

In[1095]:= Clear x, y, t

x t



1

Exp

5



y t



Exp



In[1096]:= x t



t

t



Exp t



Cos 2t



Cos 2t



Sin 2t



3 Sin 2t



y t //Simplify



Out[1096]= 0



and

In[1097]:= y t



5x t



2y t //Simplify



Out[1097]= 1



We graph this solution in several different ways. First, we graph the

solution



x



xt



parametrically with ParametricPlot in Figure 6-3.

y yt

Then, we graph x t and y t together as functions of t in Figure 6-4.

In[1098]:= ParametricPlot x t , y t , t, 0, 3Π ,

PlotRange

0 . 5, 1 ,

1, 0.5 , AspectRatio 1

Plot x t , y t , t, 0, 3Π ,

PlotStyle

GrayLevel 0 , GrayLevel 0.5

PlotRange

1 , 0. 5



,



1

Notice that limt x t

yt

0. Therefore, in the para5 and limt

metric plot, the points on the curve approach 1/ 5, 0 as t increases.



6.2 Systems of Equations: Preliminary Definitions and Theory



0.4

0.2



-0.4 -0.2



0.2



0.4



0.6



0.8



1



-0.2

-0.4

-0.6

-0.8

-1

Figure 6-3



x (on the horizontal axis) versus y (on the vertical axis)



0.4

0.2

2



4



6



8



-0.2

-0.4

-0.6

-0.8

-1

Figure 6-4 x (in black) and y (in gray) as functions of t



We will discuss techiques for solving systems in the following sections. For now,

we make the following remarks. First, we saw previously that you can often use

NDSolve to generate a numerical solution of a system, which is of particular benefit with nonlinear systems. DSolve can also often be used to find solutions of

linear systems and, in a few special cases, nonlinear systems.



431



432



Chapter 6 Systems of Ordinary Differential Equations



x



EXAMPLE 6.2.2: Solve



y



2y

1

4x



x

and



2y



y



1

4x



x0



2, y 0



1.



SOLUTION: DSolve can find a general solution of this linear system.

In[1099]:= gensol

Out[1099]=



x t



y t



DSolve x t

2y t , y t

1/ 4 x t , x t , y t , t

t

C 1 Cos

2 2 C 2 Sin

2

C 2 Cos



t



C 1 Sin



2



2



t



,



2



t

2



2



Similarly, DSolve can solve the initial-value problem. The resulting list

is named partsol.

In[1100]:= partsol



Out[1100]=



x t

y t



x t

2y t ,

1/ 4 x t , x 0

2,

1 , x t ,y t ,t

t

t

2 Cos

2 Sin

,

2

2

t

t

1

2 Cos

2 Sin

2

2

2

DSolve

y t

y 0



We use Plot to graph the x and y components of the solution individually and ParametricPlot to graph them parametrically. See

Figure 6-5.

In[1101]:= p1



p2



Plot x t , y t /.partsol,

t, 0, 4 Sqrt 2 Π ,

PlotStyle

GrayLevel 0 ,

GrayLevel 0.3 ,

AxesLabel

"t", "x, y" ,

DisplayFunction Identity

ParametricPlot x t , y t /.partsol,

t, 0, 2 Sqrt 2 Π , AxesLabel

"x", "y" ,

AspectRatio Automatic,

DisplayFunction Identity



Show GraphicsArray



p1, p2



6.2 Systems of Equations: Preliminary Definitions and Theory



433



x,y

y

1

0.5

-3 -2 -0.5

-1

-1



3

2

1

-1

-2

-3



2.5 5 7.51012.51517.5t



(a)

Figure 6-5



1



2



3



x



(b)



(a) Plots of x (in black) and y (in gray). (b) Parametric plot of x versus y

y

3

2

1

-6



-4



-2



2



4



6



x



-1

-2

-3



Figure 6-6 Direction field associated with the system together with the solution to the

initial-value problem



For an autonomous system like this, we use PlotVectorField

to graph the direction field associated with the system.

PlotVectorField is contained in the PlotField package that is located

in the Graphics folder (or directory). After loading the PlotField package,

In[1102]:= << Graphics‘PlotField‘



we use PlotVectorField to graph the direction field associated with

the system and then display the direction field together with the solution to the initial-value problem in Figure 6-6.

In[1103]:= pvf



PlotVectorField 2 y, 1/4x , x, 6, 6 ,

y, 3, 3 , DisplayFunction Identity



Show pvf, p2, PlotRange

6, 6 , 3, 3

AspectRatio Automatic,

DisplayFunction $DisplayFunction,

Axes Automatic, AxesLabel

"x", "y"



,



An autonomous system is

one for which the

independent variable does

not explicitly occur in the

equations.



434



Chapter 6 Systems of Ordinary Differential Equations



In fact, we can show the direction field together with several solutions.

With the following command, we use Map to apply a pure function to

the list {0.5,1,1.5,2,2.5} that solves the system if x 0

0 and

y0

i for i 0.5, 1.0, . . . , 2.5.



In[1104]:= severalsols

2y t ,

Map DSolve x t

y t

1/ 4 x t , x 0

2, y 0

# ,

x t , y t , t & , 0.5, 1, 1.5, 2, 2.5

Out[1104]=



x t



2. Cos



t



1.41421 Sin



2

y t



t



,



2

t



0.5 Cos



2

t



0.707107 Sin



,



2

x t



2 Cos



t

2



2 Sin



t



,



2



t

t

1

2 Cos

2 Sin

,

2

2

2

t

t

x t

2. Cos

,

4.24264 Sin

2

2

t

y t

1.5 Cos

2

t

,

0.707107 Sin

2

t

t

x t

2 Cos

,

2 2 Sin

2

2

t

t

1

4 Cos

y t

2 Sin

,

2

2

2

t

t

x t

2. Cos

7.07107 Sin

,

2

2

t

y t

2.5 Cos

2

t

0.707107 Sin

2

y t



We then use ParametricPlot to graph the solutions obtained in

severalsols together and display them with the direction field,

named pvf, in Figure 6-7.



6.2 Systems of Equations: Preliminary Definitions and Theory



435



y

3

2

1

-6



-4



-2



2



4



6



x



-1

-2

-3



Figure 6-7 Direction field associated with the system together with several solutions of

the system



In[1105]:= p3



ParametricPlot x t ,

y t /.severalsols,

t, 0, 2Sqrt 2 Π , Compiled

DisplayFunction Identity



False,



Show pvf, p3, PlotRange

6, 6 , 3, 3

AspectRatio Automatic,

DisplayFunction $DisplayFunction,

Axes Automatic, AxesLabel

"x", "y"



,



EXAMPLE 6.2.3: The Jacobi elliptic functions satisfy the nonlinear

system

du/ dt



vw



dv/ dt



uw



dw/ dt



k2 uv.



Use Mathematica to solve this system.



SOLUTION: Although Mathematica generates several error messages, we see that Mathematica is able to find a general solution of the

system, although the result is given in terms of the Jacobi elliptic function,

JacobiSN.

In[1106]:= gensol



DSolve u

v t w t

w t

u t ,v



t

,v t

u t w t ,

kˆ2 u t v t ,

t ,w t ,t



The system is nonlinear

because of the products of

the dependent variables u, v,

and w.

For this system, t is the

independent variable; u u t ,

v v t , and w w t are the

dependent variables.



436



Chapter 6 Systems of Ordinary Differential Equations



Solve

ifun Inverse functions are being used by

Solve, so some solutions may not be found.

Solve

ifun Inverse functions are being used by

Solve, so some solutions may not be found.

Solve

ifun Inverse functions are being used by

Solve, so some solutions may not be found.

General

stop Further output of

Solve

ifun will be suppressed during this calculation.



Out[1106]=



u t



2



C 1 JacobiSN



2t



C 2



2



2

v t



2C 1

2



w t



2



k C 1

C 2



,



2 C 1 JacobiSN



C 2 C 3 ,



2C 2

2



u t



C 2 C 3 ,



2t



2



k2 C 1

C 2



,



2 k2 C 1 JacobiSN



C 2 C 3 ,



2t



C 2



2



k2 C 1

C 2



C 1 JacobiSN



C 2



,



2t



C 2



2



2

v t



2C 1

2



w t



2



k C 1

C 2



,



2 C 1 JacobiSN



C 2 C 3 ,



2C 2

2



u t



C 2 C 3 ,



k2 C 1

C 2



2t



2



,



2 k2 C 1 JacobiSN



C 2 C 3 ,



k2 C 1

C 2



C 1 JacobiSN



C 2



2t



C 2



2



,

2t



C 2



2



2

v t



C 2 C 3 ,



2C 1

2



k C 1

C 2



,



2 C 1 JacobiSN



C 2 C 3 ,



k2 C 1

C 2



2t



2



,



C 2



6.2 Systems of Equations: Preliminary Definitions and Theory



Out[1106]= w t



2 k2 C 1



2C 2



2t



SuperscriptBox JacobiSN

2



2

u t



2



437



C 2



2



k C 1

C 2

C 1 JacobiSN

2t

C 2 C 3 ,



,

C 2



2



C 2 C 3 ,



2

v t



2C 1



w t



2C 2

2



,



2 C 1 JacobiSN



C 2 C 3 ,



2



k C 1

C 2



k2 C 1

C 2



k2 C 1

C 2



C 2



,



2 k2 C 1 JacobiSN



C 2 C 3 ,



2t



2



2t



C 2



2



We use the Help Browser to obtain information regarding the JacobiSN

function as indicated in the following screen shot.



438



Chapter 6 Systems of Ordinary Differential Equations



As with other equations, under reasonable conditions, a solution to a system of

differential equations always exists.

Theorem 11 (Existence and Uniqueness). Assume that each of the functions

f1 t, x1 , x2 , . . . , xn , f2 t, x1 , x2 , . . . , xn , . . . , fn t, x1 , x2 , . . . , xn

and the partial derivatives f1 / x1 , f2 / x2 , . . . , fn / xn are continuous in a region R

containing the point t0 , y1 , y2 , . . . , yn . Then, the initial-value problem

x1



f1 t, x1 , x2 , . . . , xn



x2



f2 t, x1 , x2 , . . . , xn

(6.4)



xn

x1 t0



fn t, x1 , x2 , . . . , xn

y1 , x2 t0



y2 , . . . xn t0



yn



has a unique solution



on an interval I containing t



x1



Φ1 t



x2



Φ2 t



xn



Φn t



(6.5)



t0 .



EXAMPLE 6.2.4: Show that the initial-value problem

dx/ dt

dy/ dt

x0



2x

3y

2, y 0



xy

xy

3/ 2



has a unique solution.



DSolve is not able to find

an explicit solution to this

nonlinear system.



SOLUTION: In this case, we identify f1 t, x, y

2x xy and f2 t, x, y

2 y and f2 / y

3 x. All four of these

3y xy with f1 / x

functions are continuous on a region containing 0, 2, 3/ 2 . Thus, by the

Existence and Uniqueness Theorem, a unique solution to the initialvalue problem exists. In this case, we use NDSolve to approximate the

solution to this nonlinear problem valid for 0 t 10.



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