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CHAP. 3 Higher Order Linear ODEs
PROBLEM SET 3.3
1–7
GENERAL SOLUTION
Solve the following ODEs, showing the details of your
work.
1. y t ⫹ 3y s ⫹ 3y r ⫹ y ⫽ ex ⫺ x ⫺ 1
2. y t ⫹ 2y s ⫺ y r ⫺ 2y ⫽ 1 ⫺ 4x 3
3. (D 4 ⫹ 10D 2 ⫹ 9I ) y ⫽ 6.5 sinh 2x
4. (D 3 ⫹ 3D 2 ⫺ 5D ⫺ 39I )y ⫽ ⫺300 cos x
5. (x 3D 3 ⫹ x 2D 2 ⫺ 2xD ⫹ 2I )y ⫽ x ⴚ2
6. (D 3 ⫹ 4D)y ⫽ sin x
7. (D 3 ⫺ 9D 2 ⫹ 27D ⫺ 27I )y ⫽ 27 sin 3x
8–13
INITIAL VALUE PROBLEM
Solve the given IVP, showing the details of your work.
8. y iv ⫺ 5y s ⫹ 4y ⫽ 10eⴚ3x, y(0) ⫽ 1, y r (0) ⫽ 0,
y s (0) ⫽ 0, y t (0) ⫽ 0
9. y iv ⫹ 5y s ⫹ 4y ⫽ 90 sin 4x, y(0) ⫽ 1, y r (0) ⫽ 2,
y s (0) ⫽ ⫺1, y t (0) ⫽ ⫺32
10. x 3y t ⫹ xy r ⫺ y ⫽ x 2, y(1) ⫽ 1, y r (1) ⫽ 3,
y s (1) ⫽ 14
11. (D 3 ⫺ 2D 2 ⫺ 3D)y ⫽ 74eⴚ3x sin x, y(0) ⫽ ⫺1.4,
y r (0) ⫽ 3.2, y s (0) ⫽ ⫺5.2
12. (D 3 ⫺ 2D 2 ⫺ 9D ⫹ 18I )y ⫽ e2x, y(0) ⫽ 4.5,
y r (0) ⫽ 8.8, y s (0) ⫽ 17.2
13. (D 3 ⫺ 4D)y ⫽ 10 cos x ⫹ 5 sin x, y(0) ⫽ 3,
y r (0) ⫽ ⫺2, y s (0) ⫽ ⫺1
14. CAS EXPERIMENT. Undetermined Coefficients.
Since variation of parameters is generally complicated,
it seems worthwhile to try to extend the other method.
Find out experimentally for what ODEs this is possible
and for what not. Hint: Work backward, solving ODEs
with a CAS and then looking whether the solution
could be obtained by undetermined coefficients. For
example, consider
y t ⫺ 3y s ⫹ 3y r ⫺ y ⫽ x 1>2ex
and
x 3y t ⫹ x 2y s ⫺ 2xy r ⫹ 2y ⫽ x 3 ln x.
15. WRITING REPORT. Comparison of Methods. Write
a report on the method of undetermined coefficients and
the method of variation of parameters, discussing and
comparing the advantages and disadvantages of each
method. Illustrate your findings with typical examples.
Try to show that the method of undetermined coefficients,
say, for a third-order ODE with constant coefficients and
an exponential function on the right, can be derived from
the method of variation of parameters.
CHAPTER 3 REVIEW QUESTIONS AND PROBLEMS
1. What is the superposition or linearity principle? For
what nth-order ODEs does it hold?
2. List some other basic theorems that extend from
second-order to nth-order ODEs.
3. If you know a general solution of a homogeneous linear
ODE, what do you need to obtain from it a general
solution of a corresponding nonhomogeneous linear
ODE?
4. What form does an initial value problem for an nthorder linear ODE have?
5. What is the Wronskian? What is it used for?
6–15
GENERAL SOLUTION
Solve the given ODE. Show the details of your work.
6. y iv ⫺ 3y s ⫺ 4y ⫽ 0
7. y t ⫹ 4y s ⫹ 13y r ⫽ 0
8. y t ⫺ 4y s ⫺ y r ⫹ 4y ⫽ 30e2x
9. (D 4 ⫺ 16I )y ⫽ ⫺15 cosh x
10. x 2y t ⫹ 3xy s ⫺ 2y r ⫽ 0
11. y t ⫹ 4.5y s ⫹ 6.75y r ⫹ 3.375y ⫽ 0
12. (D 3 ⫺ D)y ⫽ sinh 0.8x
13. (D 3 ⫹ 6D 2 ⫹ 12D ⫹ 8I )y ⫽ 8x 2
14. (D 4 ⫺ 13D 2 ⫹ 36I )y ⫽ 12ex
15. 4x 3y t ⫹ 3xy r ⫺ 3y ⫽ 10
INITIAL VALUE PROBLEM
16–20
Solve the IVP. Show the details of your work.
16. (D 3 ⫺ D 2 ⫺ D ⫹ I )y ⫽ 0, y(0) ⫽ 0, Dy(0) ⫽ 1,
D 2y(0) ⫽ 0
17. y t ⫹ 5y s ⫹ 24y r ⫹ 20y ⫽ x, y(0) ⫽ 1.94,
y r (0) ⫽ ⫺3.95, y s ⫽ ⫺24
18. (D 4 ⫺ 26D 2 ⫹ 25I )y ⫽ 50(x ⫹ 1)2, y(0) ⫽ 12.16,
Dy(0) ⫽ ⫺6, D 2y(0) ⫽ 34, D 3y(0) ⫽ ⫺130
19. (D 3 ⫹ 9D 2 ⫹ 23D ⫹ 15I )y ⫽ 12exp(⫺4x),
y(0) ⫽ 9, Dy(0) ⫽ ⫺41, D 2y(0) ⫽ 189
20. (D 3 ⫹ 3D 2 ⫹ 3D ⫹ I )y ⫽ 8 sin x, y(0) ⫽ ⫺1,
y r (0) ⫽ ⫺3, y s (0) ⫽ 5
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Summary of Chapter 3
123
SUMMARY OF CHAPTER
3
Higher Order Linear ODEs
Compare with the similar Summary of Chap. 2 (the case n ⴝ 2).
Chapter 3 extends Chap. 2 from order n ⫽ 2 to arbitrary order n. An nth-order
linear ODE is an ODE that can be written
(1)
y (n) ⫹ pnⴚ1(x)y (nⴚ1) ⫹ Á ⫹ p1(x)y r ⫹ p0(x)y ⫽ r (x)
with y (n) ⫽ d ny>dx n as the first term; we again call this the standard form. Equation
(1) is called homogeneous if r (x) ⬅ 0 on a given open interval I considered,
nonhomogeneous if r (x) [ 0 on I. For the homogeneous ODE
(2)
y (n) ⫹ pnⴚ1(x)y (nⴚ1) ⫹ Á ⫹ p1(x)y r ⫹ p0(x)y ⫽ 0
the superposition principle (Sec. 3.1) holds, just as in the case n ⫽ 2. A basis or
fundamental system of solutions of (2) on I consists of n linearly independent
solutions y1, Á , yn of (2) on I. A general solution of (2) on I is a linear combination
of these,
(3)
y ⫽ c1 y1 ⫹ Á ⫹ cn yn
(c1, Á , cn arbitrary constants).
A general solution of the nonhomogeneous ODE (1) on I is of the form
y ⫽ yh ⫹ yp
(4)
(Sec. 3.3).
Here, yp is a particular solution of (1) and is obtained by two methods (undetermined
coefficients or variation of parameters) explained in Sec. 3.3.
An initial value problem for (1) or (2) consists of one of these ODEs and n
initial conditions (Secs. 3.1, 3.3)
(5)
y(x 0) ⫽ K 0,
y r (x 0) ⫽ K 1,
Á,
y (nⴚ1)(x 0) ⫽ K nⴚ1
with given x 0 in I and given K 0, Á , K nⴚ1. If p0, Á , pnⴚ1, r are continuous on I,
then general solutions of (1) and (2) on I exist, and initial value problems (1), (5)
or (2), (5) have a unique solution.