Exam (elaborations)

Solutions for Differential Equations with Boundary-Value Problems, 10th Edition Zill (All Chapters included)

8 views 0 purchase

Course
Math

Institution
Math

Complete Solutions Manual for Differential Equations with Boundary-Value Problems, 10th Edition by Dennis G. Zill ; ISBN13: 9780357760451. (Full Chapters included Chapter 1 to 15)....1. INTRODUCTION TO DIFFERENTIAL EQUATIONS. 2. FIRST-ORDER DIFFERENTIAL EQUATIONS. 3. MODELING WITH FIRST-ORDER DIF...

[Show more]

Preview 4 out of 1030 pages

View example

Uploaded on May 13, 2024
Number of pages 1030
Written in 2023/2024
Type Exam (elaborations)
Contains Questions & answers

solutions
manual
answer guide
end of chapter
exercises
problems
differential
equations
boundary value problems
10th edition
10e
zill
9780357760451
solutions manual

Institution Math
Course Math

mizhouubcca

Member since 1 year 667 documents sold

CA$42.79

Added

Add to cart

Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

Differential Equations with
Boundary-Value Problems
10th Edition by Dennis G. Zill

Complete Chapter Solutions Manual
are included (Ch 1 to 15)

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

,Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With BOUNDARY VALUE PROBLEMS 2024, 9780357760451; Chapter #1:
Introduction to Diﬀerential Equations

Solution and Answer Guide
ZILL, DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS 2024,
9780357760451; 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)
p
5. Second order; nonlinear because of (dy/dx)2 or 1 + (dy/dx)2
6. Second order; nonlinear because of R2
7. Third order; linear
8. Second order; nonlinear because of ẋ2
9. First order; nonlinear because of sin (dy/dx)
10. First order; linear
11. Writing the diﬀerential equation in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear
in y because of y 2 . However, writing it in the form (y 2 − 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 y ′ = − 12 e−x/2 . Then 2y ′ + y = −e−x/2 + e−x/2 = 0.

1

,Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With BOUNDARY VALUE PROBLEMS 2024, 9780357760451; Chapter #1:
Introduction to Diﬀerential Equations

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

dy −20t 6 6 −20t
+ 20y = 24e + 20 − e = 24.
dt 5 5

15. From y = e3x cos 2x we obtain y ′ = 3e3x cos 2x−2e3x sin 2x and y ′′ = 5e3x cos 2x−12e3x sin 2x,
so that y ′′ − 6y ′ + 13y = 0.
16. From y = − cos x ln(sec x + tan x) we obtain y ′ = −1 + sin x ln(sec x + tan x) and
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 y ′ = 1+2(x+2)−1/2
we have

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

= 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 y ′ 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 6= π/2 + nπ}
or {x x 6= π/10 + nπ/5}. From y ′ = 25 sec2 5x we have

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 6= 0} or {x x 6= −2 or x 6= 2}. From y ′ =
2x/(4 − x2 )2 we have
2
1
′
y = 2x = 2xy 2 .
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 6= 0 or sin x 6= 1.
Thus, the domain is {x x =6 π/2 + 2nπ}. From y ′ = − 12 (1 − sin x)−3/2 (− cos x) we have

2y ′ = (1 − sin x)−3/2 cos x = [(1 − sin x)−1/2 ]3 cos x = y 3 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.

2

, Solution and Answer Guide: Zill, DIFFERENTIAL EQUATIONS With BOUNDARY VALUE PROBLEMS 2024, 9780357760451; Chapter #1:
Introduction to Diﬀerential Equations

21. Writing ln(2X − 1) − ln(X − 1) = t and diﬀerentiating x

implicitly we obtain 4

2 dX 1 dX
− =1 2
2X − 1 dt X − 1 dt

2 1 dX t
− =1 –4 –2 2 4
2X − 1 X − 1 dt
–2
2X − 2 − 2X + 1 dX
=1
(2X − 1) (X − 1) dt
–4
dX
= −(2X − 1)(X − 1) = (X − 1)(1 − 2X).
dt

Exponentiating both sides of the implicit solution we obtain

2X − 1
= et
X −1
2X − 1 = Xet − et

(et − 1) = (et − 2)X

et − 1
X= .
et − 2

Solving et − 2 = 0 we get t = ln 2. Thus, the solution is deﬁned on (−∞, ln 2) or on (ln 2, ∞).
The graph of the solution deﬁned on (−∞, ln 2) is dashed, and the graph of the solution
deﬁned on (ln 2, ∞) is solid.

22. Implicitly diﬀerentiating the solution, we obtain y

dy dy 4
−2x2 − 4xy + 2y =0
dx dx
2
−x2 dy − 2xy dx + y dy = 0
x
2xy dx + (x2 − y)dy = 0. –4 –2 2 4

–2
Using the quadratic formula to solve y 2 − 2x2 y − 1 = 0
√ √
for y , we get y = 2x2 ± 4x4 + 4 /2 = x2 ± x4 + 1 . –4
√
Thus, two explicit solutions are y1 = x2 + x4 + 1 and
√
y2 = x2 − x4 + 1 . Both solutions are deﬁned on (−∞, ∞).
The graph of y1 (x) is solid and the graph of y2 is dashed.

3

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller mizhouubcca. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$42.79. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75632 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling

Exam (elaborations)

Solutions for Differential Equations with Boundary-Value Problems, 10th Edition Zill (All Chapters included)

Document information

Subjects

Written for

Seller

Reviews received

Content preview