Nonhomogeneous Linear Systems of ODEs

Case III. Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example

Case 2. Damped Forced Oscillations

4.6 Nonhomogeneous Linear Systems of ODEs

13.

14. TEAM PROJECT. Self-sustained oscillations.

(a) Van der Pol equation.Determine the type of the

critical point at ( ) when .

(b) Rayleigh equation. Show that the Rayleigh equation⁵

also describes self-sustained oscillations and that by differentiating it and setting one obtains the van der Pol equation.

(c) Duffing equation.The Duffing equation is where usually is small, thus characterizing a small deviation of the restoring force from linearity.

and are called the cases of a hard spring and a soft spring, respectively. Find the equation of the trajectories in the phase plane. (Note that for all these curves are closed.)

b⬎0 b⬍0

b⬎0 ƒbƒ

ys_⫹v0²_y⫹by³_⫽0

y⫽Yr

Ys_⫺␮(1⫺ ¹_3Yr²_)Yr_{⫹Y⫽0 (␮⬎0)}

␮⬎0, ␮⫽0, ␮⬍0 0, 0

ys_{⫹sin y⫽0} 15. Trajectories. Write the ODE as a

system, solve it for as a function of , and sketch or graph some of the trajectories in the phase plane.

y₁ y₂

ys_⫺4y⫹y³_⫽0

y₂

c = 5

c = 4

c = 3

–2 2

y₁

Fig. 98. Trajectories in Problem 15

undetermined coefficients and the method of the variation of parameters; these have counterparts for a single ODE, as we know from Secs. 2.7 and 2.10.

Method of Undetermined Coefficients

Just as for a single ODE, this method is suitable if the entries of A are constants and the components of g are constants, positive integer powers of t, exponential functions, or cosines and sines. In such a case a particular solution is assumed in a form similar to g; for instance, if g has components quadratic in t, with u,v, w to be determined by substitution into (1). This is similar to Sec. 2.7, except for the Modification Rule. It suffices to show this by an example.

E X A M P L E 1 Method of Undetermined Coefficients. Modification Rule Find a general solution of

(3) .

Solution. A general equation of the homogeneous system is (see Example 1 in Sec. 4.3)

(4) .

Since is an eigenvalue of A, the function on the right side also appears in , and we must apply the Modification Rule by setting

(rather than ).

Note that the first of these two terms is the analog of the modification in Sec. 2.7, but it would not be sufficient here. (Try it.) By substitution,

.

Equating the -terms on both sides, we have . Hence u is an eigenvector of A corresponding to

; thus [see (5)] with any . Equating the other terms gives

thus .

Collecting terms and reshuffling gives

.

By addition, , and then , say, , thus,

We can simply choose . This gives the answer

(5) .

For other k we get other v; for instance, gives , so that the answerbecomes

(5*) y⫽c1 c¹ , etc. 䊏

1d ^eⴚ2t⫹c2 c ¹

⫺1d ^{eⴚ4t⫺ 2} c¹

1d ^teⴚ2t⫹ c^⫺2

2d ^eⴚ2t

v⫽[⫺2 2]^T k⫽ ⫺2

y⫽y^(h)⫹y^(p)⫽c1 c¹

1d ^{eⴚ2t⫹ c2} c ¹

⫺1d ^eⴚ4t⫺2 c¹

1d ^teⴚ2t⫹c⁰

4d ^eⴚ2t

k⫽0

v⫽[k k⫹4]^T. v₁⫽k, v₂⫽k⫹4

v₂⫽v₁⫹4 0⫽ ⫺2a⫺4, a⫽ ⫺2

⫺v₁⫹v₂⫽ ⫺a⫹2 v₁⫺v₂⫽ ⫺a⫺6

c^a

ad ^⫺c^2v_2v¹

2d ^⫽c^⫺3v_v^{1⫹ v2}

1⫺3v₂d^⫹ c^⫺6

2d

u⫺2v⫽Av⫹c^⫺6

2d

a⫽0 u⫽a[1 1]^T

l⫽ ⫺2

⫺2u⫽Au te^ⴚ2t

y^(p)r_⫽ue^ⴚ2t_⫺2ute^ⴚ2t_⫺2ve^ⴚ2t_⫽Aute^ⴚ2t_⫹Ave^ⴚ2t_⫹g

ue^ⴚ2t y^(p)⫽ute^ⴚ2t⫹ve^ⴚ2t

y^(h) e^ⴚ2t

l⫽ ⫺2

y^(h)⫽c1c¹

1d ^{eⴚ2t⫹ c2}c ¹

⫺1d ^eⴚ4t

yr_⫽Ay⫹g⫽c^{⫺3 1}

1 ⫺3d ^y⫹c^⫺6

2d ^eⴚ2t y^(p)⫽u⫹vt⫹wt²

y^(p)

SEC. 4.6 Nonhomogeneous Linear Systems of ODEs 161

Method of Variation of Parameters

This method can be applied to nonhomogeneous linear systems (6)

with variable and general . It yields a particular solution of (6) on some open interval J on the t-axis if a general solution of the homogeneous system

on Jis known. We explain the method in terms of the previous example.

E X A M P L E 2 Solution by the Method of Variation of Parameters Solve (3) in Example 1.

Solution. A basis of solutions of the homogeneous system is and . Hence the general solution (4) of the homogeneous system may be written

(7) .

Here, is the fundamental matrix (see Sec. 4.2). As in Sec. 2.10 we replace the constant vector cby a variable vector u(t) to obtain a particular solution

. Substitution into (3) gives

(8)

Now since and are solutions of the homogeneous system, we have

, , thus .

Hence , so that (8) reduces to

. The solution is ;

here we use that the inverse of Y(Sec. 4.0) exists because the determinant of Yis the Wronskian W, which is not zero for a basis. Equation (9) in Sec. 4.0 gives the form of ,

. We multiply this by g, obtaining

Integration is done componentwise (just as differentiation) and gives

(where comes from the lower limit of integration). From this and Yin (7) we obtain

. Yu⫽c^{eⴚ2t eⴚ4t}

e^ⴚ2t ⫺e^ⴚ4td c ^⫺2t

⫺2e^2t⫹2d^⫽ c^{⫺2teⴚ2t⫺2eⴚ2t⫹2eⴚ4t}

⫺2te^ⴚ2t⫹2e^ⴚ2t⫺2e^ⴚ4td ^⫽c^⫺2t⫺2

⫺2t⫹2d ^eⴚ2t⫹c ²

⫺2d ^eⴚ4t

⫹ 2

u(t)⫽

冮

₀^tc_⫺^⫺_4e^{22t~d d~t ⫽c}_⫺2e^⫺2t2t_⫹2^d

ur_⫽Y^ⴚ1_g⫽¹

2c^{e2t e2t}

e^4t ⫺e^4td c^⫺6eⴚ2t

2e^ⴚ2td ^⫽1₂c ^⫺4

⫺8e^2td ^⫽c ^⫺2

⫺4e^2td^.

Y^ⴚ1⫽ 1

⫺2e^ⴚ6tc^{⫺eⴚ4t ⫺eⴚ4t}

⫺e^ⴚ2t e^ⴚ2td ^⫽1₂c^{e2t e2t}

e^4t ⫺e^4td

Y^ⴚ1 Y^ⴚ1

ur_⫽Y^ⴚ1_g Yur_⫽g

Yr_u⫽AYu

Yr_⫽AY y⁽²⁾r_⫽Ay⁽²⁾

y⁽¹⁾r_⫽Ay⁽¹⁾ y⁽²⁾

y⁽¹⁾

Yr_u⫹Yur_⫽AYu⫹g.

yr_⫽Ay⫹g

y^(p)⫽Y(t)u(t) Y(t)⫽[y⁽¹⁾ y⁽²⁾]^T

y^(h)⫽ c^{eⴚ2t eⴚ4t}

e^ⴚ2t ⫺e^ⴚ4td c^c1

c2d ^⫽Y(t)c

[e^ⴚ4t ⫺e^ⴚ4t]^T [e^ⴚ2t e^ⴚ2t]^T

y

r

_⫽A(t)y

y^(p) g(t)

A⫽A(t)

y

r

_{⫽A(t)y⫹g(t)}

The last term on the right is a solution of the homogeneous system. Hence we can absorb it into . We thus obtain as a general solution of the system (3), in agreement with .

(9) y⫽c1 c¹ . 䊏

1d ^eⴚ2t⫹c2 c ¹

⫺1d ^eⴚ4t⫺2 c¹

1d ^teⴚ2t⫹c^⫺2

2d ^eⴚ2t

(5*)

y^(h)

SEC. 4.6 Nonhomogeneous Linear Systems of ODEs 163

1. Prove that (2) includes every solution of (1).

2–7 GENERAL SOLUTION

Find a general solution. Show the details of your work.

2.

3.

4.

5.

6.

7.

8. CAS EXPERIMENT. Undetermined Coefficients.

Find out experimentally how general you must choose , in particular when the components of g have a different form (e.g., as in Prob. 7). Write a short report, covering also the situation in the case of the modification rule.

9. Undetermined Coefficients. Explain why, in Example 1 of the text, we have some freedom in choosing the vector v.

10–15 INITIAL VALUE PROBLEM Solve, showing details:

10.

11.

12.

13.

14.

y₁(0)⫽1, y₂(0)⫽0 yr_{2⫽ ⫺y1⫺20e}^ⴚt

yr_1⫽4y2⫹5e^t

y₁(0)⫽5, y₂(0)⫽2 y₂r_{⫽ ⫺4y1⫹17 cos t}

y₁r_{⫽y2⫺5 sin t}

y₁(0)⫽2, y₂(0)⫽ ⫺1 y₂r_{⫽y1⫹y2⫺t}²_⫹t⫺1

y₁r_{⫽y1⫹4y2⫺t}²_⫹6t

y₁(0)⫽1, y₂(0)⫽0 y₂r_⫽y1⫺e^2t

y₁r_⫽y2⫹6e^2t

y₁(0)⫽19, y₂(0)⫽ ⫺23 y₂r_{⫽5y1⫹6y2⫺6e}^t

y₁r_{⫽ ⫺3y1⫺4y2⫹5e}^t

y^(p)

y₂r_{⫽5y1⫹6y2⫹3e}^ⴚt_⫺15t⫺20

yr_{1⫽ ⫺3y1⫺4y2⫹11t⫹15}

y₂r_⫽4y1⫺16t²_⫹2

y₁r_⫽4y2

y₂r_{⫽2y1⫹3y2⫺2.5t}

y₁r_{⫽4y1⫹y2⫹0.6t}

y₂r_{⫽2y1⫺6y2⫹cosh t⫹2 sinh t}

y₁r_{⫽4y1⫺8y2⫹2 cosh t}

y₂r_⫽y1⫺3e^3t

y₁r_⫽y2⫹e^3t

y₂r_{⫽3y1⫺y2⫺10 sin t}

y₁r_{⫽y1⫹y2⫹10 cos t}

15.

16. WRITING PROJECT. Undetermined Coefficients.

Write a short report in which you compare the application of the method of undetermined coefficients to a single ODE and to a system of ODEs, using ODEs and systems of your choice.

17–20 NETWORK

Find the currents in Fig. 99 (Probs. 17–19) and Fig. 100 (Prob. 20) for the following data, showing the details of your work.

17.

18. Solve Prob. 17 with and the other data as before.

19. In Prob. 17 find the particular solution when currents and charge at t⫽0are zero.

E⫽440 sin t V

E⫽200 V C⫽0.5 F,

L⫽1 H, R₂⫽8 ⍀,

R₁⫽2 ⍀,

y₁(0)⫽1, y₂(0)⫽ ⫺4 yr_{2⫽ ⫺y2⫹1⫹t}

yr_{1⫽y1⫹2y2⫹e}^2t_⫺2t

P R O B L E M S E T 4 . 6

Switch E

L

R₁ R₂

C

I₁ I₂

Fig. 99. Problems 17–19

E R₁

R₂

I₁ I₂

L₁ L₂

Fig. 100. Problem 20 20.

I₁(0)⫽I₂(0)⫽0 E⫽100 V,

L₂⫽1 H, L₁⫽0.8 H,

R₂⫽1.4 ⍀, R₁⫽1 ⍀,

1. State some applications that can be modeled by systems of ODEs.

2. What is population dynamics? Give examples.

3. How can you transform an ODE into a system of ODEs?

4. What are qualitative methods for systems? Why are they important?

5. What is the phase plane? The phase plane method? A trajectory? The phase portrait of a system of ODEs?

6. What are critical points of a system of ODEs? How did we classify them? Why are they important?

7. What are eigenvalues? What role did they play in this chapter?

8. What does stability mean in general? In connection with critical points? Why is stability important in engineering?

9. What does linearization of a system mean?

10. Review the pendulum equations and their linearizations.

11–17 GENERAL SOLUTION. CRITICAL POINTS Find a general solution. Determine the kind and stability of the critical point.

11. 12.

13. 14.

15. 16.

17.

18–19 CRITICAL POINT

What kind of critical point does have if Ahas the eigenvalues

18. 4 and 2 19.

20–23 NONHOMOGENEOUS SYSTEMS Find a general solution. Show the details of your work.

20.

21.

22.

23.

y₂r_{⫽y1⫹y2⫺cos t⫹sin t}

y₁r_{⫽y1⫹4y2⫺2 cos t}

y₂r_⫽4y1⫹y2

y₁r_{⫽y1⫹y2⫹sin t}

y₂r_⫽4y1⫹32t²

y₁r_⫽4y2

y₂r_{⫽ ⫺2y1⫺3y2⫹e}^t

y₁r_{⫽2y1⫹2y2⫹e}^t

2⫹3i, 2⫺3i

⫺

yr_⫽Ay

y₂r_{⫽ ⫺2y1⫺y2}

y₁r_{⫽ ⫺y1⫹2y2}

y₂r_{⫽ ⫺4y1}

y₂r_{⫽ ⫺2y1⫺3y2}

y₁r_⫽4y2

y₁r_{⫽ ⫺3y1⫺2y2}

y₂r_⫽3y1⫹2y2

y₂r_{⫽ ⫺y1⫺6y2}

y₁r_⫽3y1⫹4y2

y₁r_{⫽ ⫺2y1⫹5y2}

y₂r_⫽y2

y₂r_⫽8y1

y₁r_⫽5y1

y₁r_⫽2y2

24. Mixing problem. Tank in Fig. 101 initially contains 200 gal of water in which 160 lb of salt are dissolved.

Tank initially contains 100 gal of pure water. Liquid is pumped through the system as indicated, and the mixtures are kept uniform by stirring. Find the amounts of salt and in y₁(t) y₂(t) T₁and T₂, respectively.

T₂

T₁

C H A P T E R 4 R E V I E W Q U E S T I O N S A N D P R O B L E M S

T₁ Water,

10 gal/min

T₂ 6 gal/min

16 gal/min

Mixture, 10 gal/min

Fig. 101. Tanks in Problem 24

25. Network. Find the currents in Fig. 102 when , .I₂(0)⫽0

I₁(0)⫽0

E(t)⫽169 sin t V, C⫽0.04 F,

L⫽1 H, R⫽2.5 ⍀,

26. Network. Find the currents in Fig. 103 when . I₂(0)⫽1 A I₁(0)⫽1 A,

C⫽0.2 F, L⫽1.25 H,

R⫽1 ⍀,

27–30 LINEARIZATION

Find the location and kind of all critical points of the given nonlinear system by linearization.

27. 28.

29. 30.

y₂r_{⫽ ⫺8y1}

y₂r_{⫽sin y1}

y₁r_⫽2y2⫹2y2²

y₁r_{⫽ ⫺4y2}

y₂r_⫽3y1

y₂r_⫽y1⫺y1³

y₁r_{⫽cos y2}

y₁r_⫽y2

E C

L

R

I₁ I₂

Fig. 102. Network in Problem 25

C R L

I₁ I₂

Fig. 103. Network in Problem 26

Summary of Chapter 4 165

Whereas single electric circuits or single mass–spring systems are modeled by single ODEs (Chap. 2), networks of several circuits, systems of several masses and springs, and other engineering problems lead to systems of ODEs, involving several unknown functions . Of central interest are first-order systems (Sec. 4.2):

, in components,

to which higher order ODEs and systems of ODEs can be reduced (Sec. 4.1). In this summary we let , so that

(1) , in components,

Then we can represent solution curves as trajectories in the phase plane (the -plane), investigate their totality [the “phase portrait” of (1)], and study the kind and stability of the critical points (points at which both and are zero), and classify them as nodes, saddle points, centers,orspiral points(Secs. 4.3, 4.4). These phase plane methods are qualitative; with their use we can discover various general properties of solutions without actually solving the system. They are primarily used for autonomous systems, that is, systems in which tdoes not occur explicitly.

A linear systemis of the form

(2) where , , .

If , the system is called homogeneous and is of the form

(3) .

If are constants, it has solutions , where is a solution of the quadratic equation

2^{a11⫺l a12}

a₂₁ a₂₂⫺l2⫽(a₁₁⫺l)(a₂₂⫺l)⫺a₁₂a₂₁⫽0 l y⫽xe^lt

a₁₁,Á, a₂₂

y

r

_⫽Ay

g⫽0

g⫽

c

^g1

g₂

d

y⫽

c

^y1

y₂

d

A⫽

c

^{a11 a12}

a₂₁ a₂₂

d

y

r

_⫽Ay⫹g,

f₂ f₁ y₁y₂

y

r

_{1⫽f1(t, y1, y2)}

y₂

r

_{⫽f2(t, y1, y2).}

y

r

_{⫽f(t, y)}

n⫽2

y

r

_{1⫽f1(t, y1,}^Á_{, yn)}

.. .

y

r

_{n⫽fn(t, y1,}^Á_{, yn),}

y

r

_{⫽f(t, y)}

y₁(t),Á, y_n(t)

S U M M A R Y O F C H A P T E R 4

Systems of ODEs. Phase Plane. Qualitative Methods

and has components determined up to a multiplicative constant by

(These ’s are called the eigenvalues and these vectors x eigenvectors of the matrix A. Further explanation is given in Sec. 4.0.)

A system (2) with is called nonhomogeneous. Its general solution is of the form , where is a general solution of (3) and a particular solution of (2). Methods of determining the latter are discussed in Sec. 4.6.

The discussion of critical points of linear systems based on eigenvalues is summarized in Tables 4.1 and 4.2 in Sec. 4.4. It also applies to nonlinear systems if the latter are first linearized. The key theorem for this is Theorem 1 in Sec. 4.5, which also includes three famous applications, namely the pendulum and van der Pol equations and the Lotka–Volterra predator–prey population model.

yp

yh

y⫽yh⫹yp

g⫽0 l

(a11⫺l)x₁⫹a12x2⫽0.

x1, x2

x⫽0

167

C H A P T E R 5

Series Solutions of ODEs.

Special Functions

In the previous chapters, we have seen that linear ODEs with constant coefficientscan be solved by algebraic methods, and that their solutions are elementary functions known from calculus. For ODEs with variable coefficientsthe situation is more complicated, and their solutions may be nonelementary functions. Legendre’s, Bessel’s, and the hypergeometric equations are important ODEs of this kind. Since these ODEs and their solutions, the Legendre polynomials, Bessel functions, and hypergeometric functions, play an important role in engineering modeling, we shall consider the two standard methods for solving such ODEs.

The first method is called the power series method because it gives solutions in the

form of a power series .

The second method is called the Frobenius methodand generalizes the first; it gives solutions in power series, multiplied by a logarithmic term or a fractional power , in cases such as Bessel’s equation, in which the first method is not general enough.

All those more advanced solutions and various other functions not appearing in calculus are known as higher functionsor special functions, which has become a technical term.

Each of these functions is important enough to give it a name and investigate its properties and relations to other functions in great detail (take a look into Refs. [GenRef1], [GenRef10], or [All] in App. 1). Your CAS knows practically all functions you will ever need in industry or research labs, but it is up to you to find your way through this vast terrain of formulas. The present chapter may give you some help in this task.

C O M M E N T . You can study this chapter directly after Chap. 2 because it needs no material from Chaps. 3 or 4.

Prerequisite:Chap. 2.

Section that may be omitted in a shorter course:5.5.

References and Answers to Problems:App. 1 Part A, and App. 2.