Solution Manual for A First Course in Differential Equations with Modeling Applications 12th Edition by Dennis Zill

A First Course in Differential
Equations with Modeling
Applications, 12th Edition by
Dennis G. Zill




Complete Chapter Solutions Manual are
included (Ch 1 to 9)




** Immediate Download
** Swift Response
** All Chapters included

,Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With MODELING APPLICATIONS 2024, 9780357760192; Chapter #1:
Introduction to Differential Equations




Solution and Answer Guide
ZILL, DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS 2024, 9780357760192; CHAPTER
#1: INTRODUCTION TO DIFFERENTIAL EQUATIONS


TABLE OF CONTENTS
End of Section Solutions ......................................................................................................................................................................... 1
Exercises 1.1 ................................................................................................................................................................................................... 1
Exercises 1.2 ................................................................................................................................................................................................. 14
Exercises 1.3 ................................................................................................................................................................................................. 22
Chapter 1 in Review Solutions .................................................................................................................................................. 30




END OF SECTION SOLUTIONS
EXERCISES 1.1
1. Second order; linear
2. Third order; nonlinear because of (dy/dx)4
3. Fourth order; linear
4. Second order; nonlinear because of cos(r + u)
√
5. Second order; nonlinear because of (dy/dx) or 2
1 + (dy/dx)2
6. Second order; nonlinear because of R2
7. Third order; linear
8. Second order; nonlinear because of ẋ 2
9. First order; nonlinear because of sin (dy/dx)
10. First order; linear
11. Writing the differential equation in the form x(dy/dx) + y2 = 1, we see that it is nonlinear in y because of
y2. However, writing it in the form (y2 — 1)(dx/dy) + x = 0, we see that it is linear in x.
12. Writing the differential equation in the form u(dv/du) + (1 + u)v = ueu we see that it is linear in v.
However, writing it in the form (v + uv — ueu)(du/dv) + u = 0, we see that it is nonlinear in u.
13. From y = e− x/2
we obtain yj = — 1 e− x/2
. Then 2yj + y = —e− x/2
+ e− x/2 = 0.
2




1

,Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With MODELING APPLICATIONS 2024, 9780357760192; Chapter #1:
Introduction to Differential Equations


6 6
14. From y = — e—20t we obtain dy/dt = 24e−20t , so that
5 5
dy + 20y = 24e−20t 6 6 −20t
+ 20 — e = 24.
dt 5 5

15. From y = e3x cos 2x we obtain yj = 3e3x cos 2x—2e3x sin 2x and yjj = 5e3x cos 2x—12e3x sin 2x, so that yjj
— 6yj + 13y = 0.
j
16. From y = — cos x ln(sec x + tan x) we obtain y = —1 + sin x ln(sec x + tan x) and
jj jj
y = tan x + cos x ln(sec x + tan x). Then y + y = tan x.
17. The domain of the function, found by solving x+2 ≥ 0, is [—2, ∞). From yj = 1+2(x+2)−1/2
we have
j 1/2−
(y —x)y = (y — x)[1 + (2(x + 2) ]

= y — x + 2(y —x)(x + 2)−1/2

= y — x + 2[x + 4(x + 2)1/2 —x](x + 2)−1/2

= y — x + 8(x + 2)1/2(x + 2)−1/2 = y — x + 8.

An interval of definition for the solution of the differential equation is (—2, ∞) because yj is not defined at x
= —2.
18. Since tan x is not defined for x = π/2 + nπ, n an integer, the domain of y = 5 tan 5x is
{x 5x /= π/2 + nπ}
or {x x /= π/10 + nπ/5}. From y j= 25 sec 25x we have
j
y = 25(1 + tan2 5x) = 25 + 25 tan2 5x = 25 + y 2 .

An interval of definition for the solution of the differential equation is (—π/10, π/10). An- other interval is
(π/10, 3π/10), and so on.
19. The domain of the function is {x 4 — x2 /= 0} or {x x /= —2 or x /= 2}. From y j =
2x/(4 — x2)2 we have
1 2
= 2xy2.
yj = 2x
4 — x2
An interval of definition for the solution of the differential equation is (—2, 2). Other inter- vals are (—∞,
—2) and (2, ∞).
√
20. The function is y = 1/ 1 — sin x , whose domain is obtained from 1 — sin x /= 0 or sin x /= 1.
Thus, the domain is {x x /= π/2 + 2nπ}. From y j= — (11 — sin x)2 −3/2 (— cos x) we have

2yj = (1 — sin x)−3/2 cos x = [(1 — sin x)−1/2]3 cos x = y3 cos x.

An interval of definition for the solution of the differential equation is (π/2, 5π/2). Another one is (5π/2,
9π/2), and so on.




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, Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With MODELING APPLICATIONS 2024, 9780357760192; Chapter #1:
Introduction to Differential Equations




21. Writing Wln(2X W — W 1) W — W ln(X W — W 1) W W = W W t Wand Wdifferentiating x

implicitly W we W obtain 4


— = W1 2
2X W— W1 W dt X W— W1 W dt
t
2 1 dX W W – W4 –2 2 4
— = W1
2X W— W1 X W— W1 dt
–2


– W4
dX
= W—(2X W— W1)(X W— W1) W= W(X W— W1)(1 W— W2X).
dt W

Exponentiating W both W sides W of W the W implicit W solution W we W obtain

2X W— W1 W
=
t W X W—
We
W1


2X W — W1 W= WXet W — Wet

(et W — W1) W= W(et W — W2)X
et 1
X W= W .
et W — W2 W
Solving Wet W — W2 W = W 0 Wwe W get Wt W = W ln W2. W Thus, Wthe W solution Wis Wdefined W on W(—∞, Wln W2) W or Won W(ln
W2, W∞). W The W graph W of W the W solution W defined W on W (—∞, Wln W2) W is W dashed, W and W the W graph W of W the

W solution W defined W on W (ln W 2, W ∞) W is W solid.



22. Implicitly W differentiating W the W solution, W we W obtain y

2 W W dy dy 4

—2x W W — W4xy W+ W2y W = W0
dx W dx W 2
—x2 W dy W— W2xy Wdx W+ Wy Wdy W= W0
x
2xy Wdx W+ W(x2 W — Wy)dy W= W0. –4 –2 2 4


Using Wthe Wquadratic W formula W to Wsolve Wy2 W W — W 2x2y W — W 1 W W= W W 0 –2
√W √W
for Wy, Wwe Wget Wy W = 2x2 W W 4x4 W +2W4 W W /2 4W
± x + W1W.
W W= Wx –4
±
√W
Thus, Wtwo Wexplicit Wsolutions Ware Wy1 W W = W x2 x4 W + W1 W and
W
+
√W W
y2 W W = W x2 W W — x4 W + W 1 W. W Both W solutions W are W defined W on W (—∞, W∞).
The W graph W of W y1(x) W is W solid W and W the W graph W of W y2 W W is W dashed.


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