7.6 Complex Eigenvalues
C H A P T E R
7
There are many physical problems that involve a number of separate elements linked together in some manner. For example, electrical networks have this character, as do some problems in mechanics or in other fields. In these and similar cases the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first order equations. In this chapter we focus on systems of first order linear equations, utilizing some of the elementary aspects of linear algebra to unify the presentation.
7.1 Introduction
Systems of simultaneous ordinary differential equations arise naturally in problems involving several dependent variables, each of which is a function of a single independent variable. We will denote the independent variable by t, and let x , x , x , . . .
1
2
3
represent dependent variables that are functions of t. Differentiation with respect to t
will be denoted by a prime.
For example, consider the spring–mass system in Figure 7.1.1. The two masses move on a frictionless surface under the influence of external forces F (t) and F (t), 1
2
and they are also constrained by the three springs whose constants are k , k , and k , 1
2
3
339
Chapter 7. Systems of First Order Linear Equations respectively. Using arguments similar to those in Section 3.8, we find the following equations for the coordinates x and x of the two masses:
1
2
d2x
m
1 = k (x − x ) − k x + F (t) 1 dt2
2
2
1
1 1
1
= −(k + k )x + k x + F (t), 1
2
1
2 2
1
(1)
d2x
m 2 = −k x − k (x − x ) + F (t) 2 dt2
3 2
2
2
1
2
= k x − (k + k )x + F (t).
2 1
2
3
2
2
A derivation of Eqs. (1) is outlined in Problem 17.
F (t)
F (t)
1
2
k
k
k
1
2
3
m
m
1
2
x
x
1
2
FIGURE 7.1.1
A two degrees of freedom spring–mass system.
Next, consider the parallel LRC circuit shown in Figure 7.1.2. Let V be the voltage drop across the capacitor and I the current through the inductor. Then, referring to Section 3.8 and to Problem 18 of this section, we can show that the voltage and current are described by the system of equations d I = V ,
dt
L (2)
d V = − I − V ,
dt
C
RC
where L is the inductance, C the capacitance, and R the resistance.
As a final example, we mention the predator–prey problem, one of the fundamental problems of mathematical ecology, which is discussed in more detail in Let H (t) and P(t) denote the populations at time t of two species, one of which (P)
preys on the other (H ). For example, P(t) and H (t) may be the number of foxes and rabbits, respectively, in a woods, or the number of bass and redear (which are eaten by bass) in a pond. Without the prey the predators will decrease, and without the predator
C
R
LFIGURE 7.1.2 A parallel LRC circuit.
7.1
Introduction the prey will increase. A mathematical model showing how an ecological balance can be maintained when both are present was proposed about 1925 by Lotka and Volterra.
The model consists of the system of differential equations dH/dt = a H − b HP,
1
1
(3)
dP/dt = −a P + b HP, 2
2
known as the predator–prey equations. In Eqs. (3) the coefficient a is the birth rate
1
of the population H ; similarly, a is the death rate of the population P. The HP terms
2
in the two equations model the interaction of the two populations. The number of encounters between predator and prey is assumed to be proportional to the product of the populations. Since any such encounter tends to be good for the predator and bad for the prey, the sign of the HP term is negative in the first equation and positive in the second. The coefficients b and b are the coefficients of interaction between predator
1
2
and prey.
Another reason for the importance of systems of first order equations is that equations of higher order can always be transformed into such systems. This is usually required if a numerical approach is planned, because almost all codes for generating approximate numerical solutions of differential equations are written for systems of first order equations. The following example illustrates how easy it is to make the transformation.
The motion of a certain spring–mass system (see Example 3 of Section 3.8) is described E X A M P L E by the second order differential equation
1
u + 0.125u + u = 0.
(4)
Rewrite this equation as a system of first order equations.
Let x = u and x = u. Then it follows that x = x . Further, u = x . Then, by
1
2
1
2
2
substituting for u, u, and u in Eq. (4), we obtain x + 0.125x + x = 0.
2
2
1
Thus x and x satisfy the following system of two first order differential equations:
1
2
x = x , 1
2
(5)
x = −x − 0.125x .
2
1
2
The general equation of motion of a spring–mass system, mu + γ u + ku = F(t), (6)
can be transformed to a system of first order equations in the same manner. If we let x = u and x = u, and proceed as in Example 1, we quickly obtain the system
1
2
x = x , 1
2
(7)
x = −(k/m)x − (γ /m)x + F(t)/m.
2
1
2
To transform an arbitrary nth order equation y(n) = F(t, y, y, . . . , y(n−1))
(8)
Chapter 7. Systems of First Order Linear Equations into a system of n first order equations we extend the method of Example 1 by introducing the variables x , x , . . . , x defined by
1
2
n
x = y,x = y,x = y,. . . , x = y(n−1).
(9)
1
2
3
n
It then follows immediately that x = x , 1
2
x = x , 2
3
(10) ...
x
= x ,n−1
n
and from Eq. (8)
x = F(t, x , x , . . . , x ).
(11)
n
1
2
n
Equations (10) and (11) are a special case of the more general system x = F (t, x , x , . . . , x ), 1
1
1
2
n
x = F (t, x , x , . . . , x ), 2
2
1
2
n
(12) ...
x = F (t, x , x , . . . , x ).
n
n
1
2
n
In a similar way the system (1) can be reduced to a system of four first order equations of the form (12), while the systems (2) and (3) are already in this form. In fact, systems of the form (12) include almost all cases of interest, so much of the more advanced theory of differential equations is devoted to such systems.
The system (12) is said to have a the interval I : α < t < β if there exists a set of n functions x = φ (t),x = φ (t),. . . , x = φ (t), (13)
1
1
2
2
n
n
that are differentiable at all points in the interval I and that satisfy the system of equations (12) at all points in this interval. In addition to the given system of differential equations there may also be given initial conditions of the form x (t ) = x0,x (t ) = x0,. . . , x (t ) = x0,
(14)
1
0
1
2
0
2
n
0
n
where t is a specified value of t in I , and x0, . . . , x0 are prescribed numbers. The
0
1
n
differential equations (12) and initial conditions (14) together form an initial value problem.
A solution (13) can be viewed as a set of parametric equations in an n-dimensional space. For a given value of t, Eqs. (13) give values for the coordinates x , . . . , x of a
1
n
point in the space. As t changes, the coordinates in general also change. The collection of points corresponding to α < t < β form a curve in the space. It is often helpful to think of the curve as the trajectory or path of a particle moving in accordance with the system of differential equations (12). The initial conditions (14) determine the starting point of the moving particle.
The following conditions on F , F , . . . , F , which are easily checked in specific
1
2
n
problems, are sufficient to assure that the initial value problem (12), (14) has a unique solution. Theorem 7.1.1 is analogous to Theorem 2.4.2, the existence and uniqueness theorem for a single first order equation.
7.1
Introduction
Theorem 7.1.1 Let each of the functions F , . . . , F and the partial derivatives ∂ F /∂ x , . . . ,
1
n
1
1
∂ F /∂x , . . . , ∂ F /∂x , . . . , ∂ F /∂x be continuous in a region R of tx x · · · x 1
n
n
1
n
n
1 2
n
space defined by α < t < β, α < x < β , . . . , α < x < β , and let the point
1
1
1
n
n
n(t , x0, x0, . . . , x0) be in R. Then there is an interval |t − t | < h in which there
0
1
2
n
0
exists a unique solution x = φ (t), . . . , x = φ (t) of the system of differential
1
1
n
n
equations (12) that also satisfies the initial conditions (14).
The proof of this theorem can be constructed by generalizing the argument in Section 2.8, but we do not give it here. However, note that in the hypotheses of the theorem nothing is said about the partial derivatives of F , . . . , F with respect to the inde 1
n
pendent variable t. Also, in the conclusion, the length 2h of the interval in which the solution exists is not specified exactly, and in some cases may be very short. Finally, the same result can be established on the basis of somewhat weaker, but more complicated hypotheses, so the theorem as stated is not the most general one known, and the given conditions are sufficient, but not necessary, for the conclusion to hold.
If each of the functions F , . . . , F in Eqs. (12) is a linear function of the dependent
1
n
variables x , . . . , x , then the system of equations is said to otherwise, it is
1
n
hus the most general system of n first order linear equations has the form x = p (t)x + · · · + p (t)x + g (t),
1
11
1
1n
n
1
x = p (t)x + · · · + p (t)x + g (t),
2
21
1
2n
n
2
(15) ...
x = p (t)x + · · · + p (t)x + g (t).
n
n1
1
nn
n
n
If each of the functions g (t), . . . , g (t) is zero for all t in the interval I , then the system
1
n
(15) is said to be otherwise, it is Observe that the systems (1) and (2) are both linear, but the system (3) is nonlinear. The is nonhomogeneous unless F (t) = F (t) = 0, while the system (2) is homogeneous.
1
2
For the linear system (15) the existence and uniqueness theorem is simpler and also has a stronger conclusion. It is analogous to Theorems 2.4.1 and 3.2.1.
Theorem 7.1.2
If the functions p , p , . . . , p , g , . . . , g are continuous on an open interval
11
12
nn
1
n
I : α < t < β, then there exists a unique solution x = φ (t), . . . , x = φ (t) of the
1
1
n
n
system (15) that also satisfies the initial conditions (14), where t is any point in I
0
and x0, . . . , x0 are any prescribed numbers. Moreover, the solution exists throughout
1
n the interval I .
Note that in contrast to the situation for a nonlinear system the existence and unique ness of the solution of a linear system are guaranteed throughout the interval in which the hypotheses are satisfied. Furthermore, for a linear system the initial values x0, . . . , x0
1
n
at t = t are completely arbitrary, whereas in the nonlinear case the initial point must
0
lie in the region R defined in Theorem 7.1.1.
The rest of this chapter is devoted to systems of linear first order equations (nonlinear systems are included in the discussion in Chapters 8 and 9). Our presentation makes use of matrix notation and assumes that you have some familiarity with the properties of matrices. The basic facts about matrices are summarized in Sections 7.2 and 7.3 and some more advanced material is reviewed as needed in later sections.
Chapter 7. Systems of First Order Linear Equations
PROBLEMS
In each of Problems 1 through 4 transform the given equation into a system of first order equations.
0
0
Transform this problem into an initial value problem for two first order equations.
7. Systems of first order equations can sometimes be transformed into a single equation of higher order. Consider the system
1
1
2
2
1
2
2
1
1
2
1
2
1
2
obtained in part (b).
1
2
1 2
1
1
2
1
1
1
2
1
2
1
2
2
2
2
1
2
2
1
1
2
1
1
2
1
2
1
2
2
2
1
2
1
1
2
1
2
1
2
2
not both zero, and if the
11
12
21
22
12
21
1
2
1
11 1
12 2
1
1
1
2
21 1
22 2
2
2
2
can be transformed into an initial value problem for a single second order equation. Can the same procedure be carried out if a , . . . , a are functions of t?
11
22
15. Consider the linear homogeneous system x = p (t)x + p (t)y,
11
12
y = p (t)x + p (t)y.
21
22
Show that if x = x (t), y = y (t) and x = x (t), y = y (t) are two solutions of the given
1
1
2
2
system, then x = c x (t) + c x (t), y = c y (t) + c y (t) is also a solution for any con 1 1
2 2
1 1
2 2 stants c and c . This is the principle of superposition.
1
2
16. Let x = x (t), y = y (t) and x = x (t), y = y (t) be any two solutions of the linear
1
1
2
2
nonhomogeneous system x = p (t)x + p (t)y + g (t), 11
12
1
y = p (t)x + p (t)y + g (t).
21
22
2
7.1
Introduction
345
Show that x = x (t) − x (t), y = y (t) − y (t) is a solution of the corresponding homo 1
2
1
2
geneous system.
17. Equations (1) can be derived by drawing a free-body diagram showing the forces acting on each mass. Figure 7.1.3a shows the situation when the displacements x and x of the two
1
2
masses are both positive (to the right) and x > x . Then springs 1 and 2 are elongated and
2
1
spring 3 is compressed, giving rise to forces as shown in Figure 7.1.3b. Use Newton’s law (F = ma) to derive Eqs. (1).
k
k
k
1
2
3
m
m
1
2
x
x
1
2
(a) F (t)
F (t)
1
k xk (x – x )
2
1 1
2
2
1
m
m
1
2
k (x – x ) k x
2
2
1
3 2 (b)
FIGURE 7.1.3 (a) The displacements x and x are both positive. (b) The free-body diagram
1
2
for the spring–mass system.
Electric Circuits.
The theory of electric circuits, such as that shown in Figure 7.1.2, consisting of inductors, resistors, and capacitors, is based on Kirchhoff’s laws: (1) The net flow of current through each node (or junction) is zero, (2) the net voltage drop around each closed loop is zero. In addition to Kirchhoff’s laws we also have the relation between the current I in amperes through each circuit element and the voltage drop V in volts across the element; namely, V = R I,R = resistance in ohms; d V
C
= I,C = capacitance in farads1;
dt
d I
L
= V,L = inductance in henrys.
dt
Kirchhoff’s laws and the current–voltage relation for each circuit element provide a system of algebraic and differential equations from which the voltage and current throughout the circuit can be determined. Problems 18 through 20 illustrate the procedure just described.
18. Consider the circuit shown in Figure 7.1.2. Let I , I , and I be the current through
1
2
3
the capacitor, resistor, and inductor, respectively. Likewise, let V , V , and V be the
1
2
3
corresponding voltage drops. The arrows denote the arbitrarily chosen directions in which currents and voltage drops will be taken to be positive.
(a) Applying Kirchhoff’s second law to the upper loop in the circuit, show that V − V = 0.
(i)
1
2
In a similar way, show that V − V = 0.
(ii)
2
3
1Actual capacitors typically have capacitances measured in microfarads. We use farad as the unit for numerical convenience.
Chapter 7. Systems of First Order Linear Equations (b) Applying Kirchhoff’s first law to either node in the circuit, show that I + I + I = 0.
(iii)
1
2
3
(c) Use the current–voltage relation through each element in the circuit to obtain the equations C V = I ,V = R I ,L I = V .
(iv)
1
1
2
2
3
3
(d) Eliminate V , V , I , and I among Eqs. (i) through (iv) to obtain
2
3
1
2
VC V = −I − 1 ,L I = V .
(v)
1
3
R
3
1
Observe that if we omit the subscripts in Eqs. (v), then we have the system (2) of the text.
19. Consider the circuit shown in Figure 7.1.4. Use the method outlined in Problem 18 to show d I = −I − V,d V = 2I − V.
dt
dt
R = 1 ohm
L = 1 henry
R = 2 ohms
C = farad
1
2
FIGURE 7.1.4 The circuit in Problem 19.
20. Consider the circuit shown in Figure 7.1.5. Use the method outlined in Problem 18 to show that the current I through the inductor and the voltage V across the capacitor satisfy the system of differential equations d Id V
L
= −R I − V,
C = I − V .
dt
1
dt
R2
L
R
R
1
2
CFIGURE 7.1.5 The circuit in Problem 20.
21. Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1 initially contains 30 gal of water and 25 oz of salt, while Tank 2 initially contains 20 gal of water and 15 oz
7.1
Introduction
1.5 gal/min
1 gal/min
1 oz/gal
3 oz/gal
3 gal/min
Q (t) oz salt Q (t) oz salt
1
2
30 gal water 20 gal water
1.5 gal/min
Tank 1
Tank 2
2.5 gal/min
FIGURE 7.1.6
Two interconnected tanks (Problem 21).
1
2
1
2
1
2
will approach its equilibrium state more rapidly?
1
1
1
2
2
2
1
2
1
2
22. Consider two interconnected tanks similar to those in Figure 7.1.6. Tank 1 initially contains
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
7
There are many physical problems that involve a number of separate elements linked together in some manner. For example, electrical networks have this character, as do some problems in mechanics or in other fields. In these and similar cases the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first order equations. In this chapter we focus on systems of first order linear equations, utilizing some of the elementary aspects of linear algebra to unify the presentation.
7.1 Introduction
Systems of simultaneous ordinary differential equations arise naturally in problems involving several dependent variables, each of which is a function of a single independent variable. We will denote the independent variable by t, and let x , x , x , . . .
1
2
3
represent dependent variables that are functions of t. Differentiation with respect to t
will be denoted by a prime.
For example, consider the spring–mass system in Figure 7.1.1. The two masses move on a frictionless surface under the influence of external forces F (t) and F (t), 1
2
and they are also constrained by the three springs whose constants are k , k , and k , 1
2
3
339
Chapter 7. Systems of First Order Linear Equations respectively. Using arguments similar to those in Section 3.8, we find the following equations for the coordinates x and x of the two masses:
1
2
d2x
m
1 = k (x − x ) − k x + F (t) 1 dt2
2
2
1
1 1
1
= −(k + k )x + k x + F (t), 1
2
1
2 2
1
(1)
d2x
m 2 = −k x − k (x − x ) + F (t) 2 dt2
3 2
2
2
1
2
= k x − (k + k )x + F (t).
2 1
2
3
2
2
A derivation of Eqs. (1) is outlined in Problem 17.
F (t)
F (t)
1
2
k
k
k
1
2
3
m
m
1
2
x
x
1
2
FIGURE 7.1.1
A two degrees of freedom spring–mass system.
Next, consider the parallel LRC circuit shown in Figure 7.1.2. Let V be the voltage drop across the capacitor and I the current through the inductor. Then, referring to Section 3.8 and to Problem 18 of this section, we can show that the voltage and current are described by the system of equations d I = V ,
dt
L (2)
d V = − I − V ,
dt
C
RC
where L is the inductance, C the capacitance, and R the resistance.
As a final example, we mention the predator–prey problem, one of the fundamental problems of mathematical ecology, which is discussed in more detail in Let H (t) and P(t) denote the populations at time t of two species, one of which (P)
preys on the other (H ). For example, P(t) and H (t) may be the number of foxes and rabbits, respectively, in a woods, or the number of bass and redear (which are eaten by bass) in a pond. Without the prey the predators will decrease, and without the predator
C
R
LFIGURE 7.1.2 A parallel LRC circuit.
7.1
Introduction the prey will increase. A mathematical model showing how an ecological balance can be maintained when both are present was proposed about 1925 by Lotka and Volterra.
The model consists of the system of differential equations dH/dt = a H − b HP,
1
1
(3)
dP/dt = −a P + b HP, 2
2
known as the predator–prey equations. In Eqs. (3) the coefficient a is the birth rate
1
of the population H ; similarly, a is the death rate of the population P. The HP terms
2
in the two equations model the interaction of the two populations. The number of encounters between predator and prey is assumed to be proportional to the product of the populations. Since any such encounter tends to be good for the predator and bad for the prey, the sign of the HP term is negative in the first equation and positive in the second. The coefficients b and b are the coefficients of interaction between predator
1
2
and prey.
Another reason for the importance of systems of first order equations is that equations of higher order can always be transformed into such systems. This is usually required if a numerical approach is planned, because almost all codes for generating approximate numerical solutions of differential equations are written for systems of first order equations. The following example illustrates how easy it is to make the transformation.
The motion of a certain spring–mass system (see Example 3 of Section 3.8) is described E X A M P L E by the second order differential equation
1
u + 0.125u + u = 0.
(4)
Rewrite this equation as a system of first order equations.
Let x = u and x = u. Then it follows that x = x . Further, u = x . Then, by
1
2
1
2
2
substituting for u, u, and u in Eq. (4), we obtain x + 0.125x + x = 0.
2
2
1
Thus x and x satisfy the following system of two first order differential equations:
1
2
x = x , 1
2
(5)
x = −x − 0.125x .
2
1
2
The general equation of motion of a spring–mass system, mu + γ u + ku = F(t), (6)
can be transformed to a system of first order equations in the same manner. If we let x = u and x = u, and proceed as in Example 1, we quickly obtain the system
1
2
x = x , 1
2
(7)
x = −(k/m)x − (γ /m)x + F(t)/m.
2
1
2
To transform an arbitrary nth order equation y(n) = F(t, y, y, . . . , y(n−1))
(8)
Chapter 7. Systems of First Order Linear Equations into a system of n first order equations we extend the method of Example 1 by introducing the variables x , x , . . . , x defined by
1
2
n
x = y,x = y,x = y,. . . , x = y(n−1).
(9)
1
2
3
n
It then follows immediately that x = x , 1
2
x = x , 2
3
(10) ...
x
= x ,n−1
n
and from Eq. (8)
x = F(t, x , x , . . . , x ).
(11)
n
1
2
n
Equations (10) and (11) are a special case of the more general system x = F (t, x , x , . . . , x ), 1
1
1
2
n
x = F (t, x , x , . . . , x ), 2
2
1
2
n
(12) ...
x = F (t, x , x , . . . , x ).
n
n
1
2
n
In a similar way the system (1) can be reduced to a system of four first order equations of the form (12), while the systems (2) and (3) are already in this form. In fact, systems of the form (12) include almost all cases of interest, so much of the more advanced theory of differential equations is devoted to such systems.
The system (12) is said to have a the interval I : α < t < β if there exists a set of n functions x = φ (t),x = φ (t),. . . , x = φ (t), (13)
1
1
2
2
n
n
that are differentiable at all points in the interval I and that satisfy the system of equations (12) at all points in this interval. In addition to the given system of differential equations there may also be given initial conditions of the form x (t ) = x0,x (t ) = x0,. . . , x (t ) = x0,
(14)
1
0
1
2
0
2
n
0
n
where t is a specified value of t in I , and x0, . . . , x0 are prescribed numbers. The
0
1
n
differential equations (12) and initial conditions (14) together form an initial value problem.
A solution (13) can be viewed as a set of parametric equations in an n-dimensional space. For a given value of t, Eqs. (13) give values for the coordinates x , . . . , x of a
1
n
point in the space. As t changes, the coordinates in general also change. The collection of points corresponding to α < t < β form a curve in the space. It is often helpful to think of the curve as the trajectory or path of a particle moving in accordance with the system of differential equations (12). The initial conditions (14) determine the starting point of the moving particle.
The following conditions on F , F , . . . , F , which are easily checked in specific
1
2
n
problems, are sufficient to assure that the initial value problem (12), (14) has a unique solution. Theorem 7.1.1 is analogous to Theorem 2.4.2, the existence and uniqueness theorem for a single first order equation.
7.1
Introduction
Theorem 7.1.1 Let each of the functions F , . . . , F and the partial derivatives ∂ F /∂ x , . . . ,
1
n
1
1
∂ F /∂x , . . . , ∂ F /∂x , . . . , ∂ F /∂x be continuous in a region R of tx x · · · x 1
n
n
1
n
n
1 2
n
space defined by α < t < β, α < x < β , . . . , α < x < β , and let the point
1
1
1
n
n
n(t , x0, x0, . . . , x0) be in R. Then there is an interval |t − t | < h in which there
0
1
2
n
0
exists a unique solution x = φ (t), . . . , x = φ (t) of the system of differential
1
1
n
n
equations (12) that also satisfies the initial conditions (14).
The proof of this theorem can be constructed by generalizing the argument in Section 2.8, but we do not give it here. However, note that in the hypotheses of the theorem nothing is said about the partial derivatives of F , . . . , F with respect to the inde 1
n
pendent variable t. Also, in the conclusion, the length 2h of the interval in which the solution exists is not specified exactly, and in some cases may be very short. Finally, the same result can be established on the basis of somewhat weaker, but more complicated hypotheses, so the theorem as stated is not the most general one known, and the given conditions are sufficient, but not necessary, for the conclusion to hold.
If each of the functions F , . . . , F in Eqs. (12) is a linear function of the dependent
1
n
variables x , . . . , x , then the system of equations is said to otherwise, it is
1
n
hus the most general system of n first order linear equations has the form x = p (t)x + · · · + p (t)x + g (t),
1
11
1
1n
n
1
x = p (t)x + · · · + p (t)x + g (t),
2
21
1
2n
n
2
(15) ...
x = p (t)x + · · · + p (t)x + g (t).
n
n1
1
nn
n
n
If each of the functions g (t), . . . , g (t) is zero for all t in the interval I , then the system
1
n
(15) is said to be otherwise, it is Observe that the systems (1) and (2) are both linear, but the system (3) is nonlinear. The is nonhomogeneous unless F (t) = F (t) = 0, while the system (2) is homogeneous.
1
2
For the linear system (15) the existence and uniqueness theorem is simpler and also has a stronger conclusion. It is analogous to Theorems 2.4.1 and 3.2.1.
Theorem 7.1.2
If the functions p , p , . . . , p , g , . . . , g are continuous on an open interval
11
12
nn
1
n
I : α < t < β, then there exists a unique solution x = φ (t), . . . , x = φ (t) of the
1
1
n
n
system (15) that also satisfies the initial conditions (14), where t is any point in I
0
and x0, . . . , x0 are any prescribed numbers. Moreover, the solution exists throughout
1
n the interval I .
Note that in contrast to the situation for a nonlinear system the existence and unique ness of the solution of a linear system are guaranteed throughout the interval in which the hypotheses are satisfied. Furthermore, for a linear system the initial values x0, . . . , x0
1
n
at t = t are completely arbitrary, whereas in the nonlinear case the initial point must
0
lie in the region R defined in Theorem 7.1.1.
The rest of this chapter is devoted to systems of linear first order equations (nonlinear systems are included in the discussion in Chapters 8 and 9). Our presentation makes use of matrix notation and assumes that you have some familiarity with the properties of matrices. The basic facts about matrices are summarized in Sections 7.2 and 7.3 and some more advanced material is reviewed as needed in later sections.
Chapter 7. Systems of First Order Linear Equations
PROBLEMS
In each of Problems 1 through 4 transform the given equation into a system of first order equations.
0
0
Transform this problem into an initial value problem for two first order equations.
7. Systems of first order equations can sometimes be transformed into a single equation of higher order. Consider the system
1
1
2
2
1
2
2
1
1
2
1
2
1
2
obtained in part (b).
1
2
1 2
1
1
2
1
1
1
2
1
2
1
2
2
2
2
1
2
2
1
1
2
1
1
2
1
2
1
2
2
2
1
2
1
1
2
1
2
1
2
2
not both zero, and if the
11
12
21
22
12
21
1
2
1
11 1
12 2
1
1
1
2
21 1
22 2
2
2
2
can be transformed into an initial value problem for a single second order equation. Can the same procedure be carried out if a , . . . , a are functions of t?
11
22
15. Consider the linear homogeneous system x = p (t)x + p (t)y,
11
12
y = p (t)x + p (t)y.
21
22
Show that if x = x (t), y = y (t) and x = x (t), y = y (t) are two solutions of the given
1
1
2
2
system, then x = c x (t) + c x (t), y = c y (t) + c y (t) is also a solution for any con 1 1
2 2
1 1
2 2 stants c and c . This is the principle of superposition.
1
2
16. Let x = x (t), y = y (t) and x = x (t), y = y (t) be any two solutions of the linear
1
1
2
2
nonhomogeneous system x = p (t)x + p (t)y + g (t), 11
12
1
y = p (t)x + p (t)y + g (t).
21
22
2
7.1
Introduction
345
Show that x = x (t) − x (t), y = y (t) − y (t) is a solution of the corresponding homo 1
2
1
2
geneous system.
17. Equations (1) can be derived by drawing a free-body diagram showing the forces acting on each mass. Figure 7.1.3a shows the situation when the displacements x and x of the two
1
2
masses are both positive (to the right) and x > x . Then springs 1 and 2 are elongated and
2
1
spring 3 is compressed, giving rise to forces as shown in Figure 7.1.3b. Use Newton’s law (F = ma) to derive Eqs. (1).
k
k
k
1
2
3
m
m
1
2
x
x
1
2
(a) F (t)
F (t)
1
k xk (x – x )
2
1 1
2
2
1
m
m
1
2
k (x – x ) k x
2
2
1
3 2 (b)
FIGURE 7.1.3 (a) The displacements x and x are both positive. (b) The free-body diagram
1
2
for the spring–mass system.
Electric Circuits.
The theory of electric circuits, such as that shown in Figure 7.1.2, consisting of inductors, resistors, and capacitors, is based on Kirchhoff’s laws: (1) The net flow of current through each node (or junction) is zero, (2) the net voltage drop around each closed loop is zero. In addition to Kirchhoff’s laws we also have the relation between the current I in amperes through each circuit element and the voltage drop V in volts across the element; namely, V = R I,R = resistance in ohms; d V
C
= I,C = capacitance in farads1;
dt
d I
L
= V,L = inductance in henrys.
dt
Kirchhoff’s laws and the current–voltage relation for each circuit element provide a system of algebraic and differential equations from which the voltage and current throughout the circuit can be determined. Problems 18 through 20 illustrate the procedure just described.
18. Consider the circuit shown in Figure 7.1.2. Let I , I , and I be the current through
1
2
3
the capacitor, resistor, and inductor, respectively. Likewise, let V , V , and V be the
1
2
3
corresponding voltage drops. The arrows denote the arbitrarily chosen directions in which currents and voltage drops will be taken to be positive.
(a) Applying Kirchhoff’s second law to the upper loop in the circuit, show that V − V = 0.
(i)
1
2
In a similar way, show that V − V = 0.
(ii)
2
3
1Actual capacitors typically have capacitances measured in microfarads. We use farad as the unit for numerical convenience.
Chapter 7. Systems of First Order Linear Equations (b) Applying Kirchhoff’s first law to either node in the circuit, show that I + I + I = 0.
(iii)
1
2
3
(c) Use the current–voltage relation through each element in the circuit to obtain the equations C V = I ,V = R I ,L I = V .
(iv)
1
1
2
2
3
3
(d) Eliminate V , V , I , and I among Eqs. (i) through (iv) to obtain
2
3
1
2
VC V = −I − 1 ,L I = V .
(v)
1
3
R
3
1
Observe that if we omit the subscripts in Eqs. (v), then we have the system (2) of the text.
19. Consider the circuit shown in Figure 7.1.4. Use the method outlined in Problem 18 to show d I = −I − V,d V = 2I − V.
dt
dt
R = 1 ohm
L = 1 henry
R = 2 ohms
C = farad
1
2
FIGURE 7.1.4 The circuit in Problem 19.
20. Consider the circuit shown in Figure 7.1.5. Use the method outlined in Problem 18 to show that the current I through the inductor and the voltage V across the capacitor satisfy the system of differential equations d Id V
L
= −R I − V,
C = I − V .
dt
1
dt
R2
L
R
R
1
2
CFIGURE 7.1.5 The circuit in Problem 20.
21. Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1 initially contains 30 gal of water and 25 oz of salt, while Tank 2 initially contains 20 gal of water and 15 oz
7.1
Introduction
1.5 gal/min
1 gal/min
1 oz/gal
3 oz/gal
3 gal/min
Q (t) oz salt Q (t) oz salt
1
2
30 gal water 20 gal water
1.5 gal/min
Tank 1
Tank 2
2.5 gal/min
FIGURE 7.1.6
Two interconnected tanks (Problem 21).
1
2
1
2
1
2
will approach its equilibrium state more rapidly?
1
1
1
2
2
2
1
2
1
2
22. Consider two interconnected tanks similar to those in Figure 7.1.6. Tank 1 initially contains
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2