Solutions for Differential Equations and Linear Algebra, 4th edition by Edwards

SOLUTIONS for
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Differential Equations and Linear
Algebra, 4th edition
Author (s): Henry C. Edwards, David E. Penney
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, CHAPTER 1
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FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1
DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS
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The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of dif-
ferential equations, and to show the student what is meant by a solution of a differential equation.
Also, the use of differential equations in the mathematical modeling of real-world phenomena is
outlined.
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Problems 1-12 are routine verifications by direct substitution of the suggested solutions into the
given differential equations. We include here just some typical examples of such verifications.

3. If y1  cos 2 x and y2  sin 2 x , then y1   2sin 2 x y2  2 cos 2 x , so
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y1  4 cos 2 x  4 y1 and y2  4sin 2 x  4 y2 . Thus y1  4 y1  0 and y2  4 y2  0 .

4. If y1  e 3 x and y 2  e 3 x , then y1  3 e3 x and y2   3 e 3 x , so y1  9e 3 x  9 y1 and
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y2  9e 3 x  9 y2 .


5. If y  e x  e x , then y  e x  e  x , so y   y   e x  e  x    e x  e  x   2 e  x . Thus
y  y  2 e x .
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6. If y1  e 2 x and y2  x e 2 x , then y1   2 e 2 x , y1  4 e 2 x , y 2  e 2 x  2 x e 2 x , and
y 2   4 e 2 x  4 x e 2 x . Hence
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y1  4 y1  4 y1   4 e 2 x   4  2 e 2 x   4  e 2 x   0
and
  4e  4 x e 2 x   4  e 2 x  2 x e 2 x   4  x e 2 x   0.
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y2  4 y2  4 y2  2 x




8. If y1  cos x  cos 2 x and y2  sin x  cos 2 x , then y1   sin x  2sin 2 x,
y1   cos x  4 cos 2 x, y2  cos x  2sin 2 x , and y2   sin x  4 cos 2 x. Hence
y1  y1    cos x  4 cos 2 x    cos x  cos 2 x   3cos 2 x
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and
y2  y2    sin x  4cos 2 x    sin x  cos 2 x   3cos 2 x.
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, 2 Chapter 1: First-Order Differential Equations


11. If y  y1  x 2 , then y   2 x 3 and y  6 x 4 , so
x 2 y   5 x y   4 y  x 2  6 x 4   5 x  2 x 3   4  x 2   0.
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If y  y 2  x 2 ln x , then y  x 3  2 x 3 ln x and y   5 x 4  6 x 4 ln x , so
x 2 y   5 x y  4 y  x 2  5 x 4  6 x 4 ln x   5 x  x 3  2 x 3 ln x   4  x 2 ln x 
  5 x 2  5 x 2    6 x 2  10 x 2  4 x 2  ln x  0.
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13. Substitution of y  erx into 3 y   2 y gives the equation 3r erx  2 erx , which simplifies
to 3 r  2. Thus r  .

14. Substitution of y  erx into 4 y  y gives the equation 4r 2 e rx  e rx , which simplifies to
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4 r 2  1. Thus r   .

15. Substitution of y  erx into y   y   2 y  0 gives the equation r 2 e rx  r e rx  2 e rx  0 ,
which simplifies to r 2  r  2  (r  2)(r  1)  0. Thus r  2 or r  1 .
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16. Substitution of y  erx into 3 y   3 y   4 y  0 gives the equation
3r 2e rx  3r e rx  4 e rx  0, which simplifies to 3r 2  3r  4  0 . The quadratic formula then
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gives the solutions r  3  57   6.

The verifications of the suggested solutions in Problems 17-26 are similar to those in Problems
1-12. We illustrate the determination of the value of C only in some typical cases. However, we
illustrate typical solution curves for each of these problems.
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17. C2 18. C 3
Problem 17 Problem 18
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4 5


(0, 3)
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(0, 2)



y y
0 0
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−4 −5
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−4 0 4 −5 0 5
x x


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, Section 1.1: Differential Equations and Mathematical Models 3


19. If y  x   Ce x  1 , then y  0   5 gives C  1  5 , so C  6 .

If y  x   C e x  x  1 , then y  0   10 gives C  1  10 , or C  11 .
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20.

Problem 19 Problem 20
10 20



5 (0, 5) (0, 10)
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y y
0 0
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−5



−10 −20
−5 0 5 −10 −5 0 5 10
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x x

21. C  7.
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22. If y ( x)  ln  x  C  , then y  0   0 gives ln C  0 , so C  1 .

Problem 21 Problem 22
10 5
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(0, 7)

5
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y y
0 0
(0, 0)
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−5



−10 −5
−2 −1 0 1 2 −20 −10 0 10 20
x x
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23. If y ( x )  14 x 5  C x 2 , then y  2  1 gives 14  32  C  81  1 , or C  56 .

C  17 .
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