Its a Solution Manual for Differential Equations and Boundary Value Problems: Computing and Modeling by C. Edwards, David Penney, David Calvis, 6th Edition. All chapters covered.

Solution Manual
Differential Equations and Boundary Value Problems:
Computing and Modeling, 6th Edition.
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INSTRUCTOR’S
SOLUTIONS MANUAL


DIFFERENTIAL EQUATIONS
AND BOUNDARY VALUE PROBLEMS
COMPUTING AND MODELING
SIXTH EDITION
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C. Henry Edwards
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David E. Penney
The University of Georgia
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David Calvis
Baldwin-Wallace University
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Boston Columbus Indianapolis New York San Francisco Hoboken
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,CHAPTER 1

FIRST-ORDER DIFFERENTIAL EQUATIONS
SECTION 1.1
DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS

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.

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
y1  4 cos 2 x  4 y1 and y2  4sin 2 x  4 y2 . Thus y1  4 y1  0 and y2  4 y2  0 .
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4. If y1  e3 x and y2  e 3 x , then y1  3 e3 x and y2   3 e 3 x , so y1  9e3 x  9 y1 and
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
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y  y  2 e  x .

6. If y1  e 2 x and y2  x e 2 x , then y1   2 e 2 x , y1  4 e 2 x , y2  e 2 x  2 x e 2 x , and
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y2   4 e 2 x  4 x e 2 x . Hence
y1  4 y1  4 y1   4 e 2 x   4  2 e 2 x   4  e 2 x   0
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and
y2  4 y2  4 y2    4e 2 x
 4 x e 2 x   4  e 2 x  2 x e 2 x   4  x e 2 x   0.
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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
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y1  y1    cos x  4 cos 2 x    cos x  cos 2 x   3cos 2 x
and
y2  y2    sin x  4 cos 2 x    sin x  cos 2 x   3cos 2 x.
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, 2 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELS


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.

If y  y2  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.


13. Substitution of y  erx into 3 y   2 y gives the equation 3r e rx  2 e rx , 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
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 .

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

The verifications of the suggested solutions in Problems 17-26 are similar to those in Problems
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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.

17. C2 18. C 3
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