dy_ vy – v,V + y² dx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
I need to do the exercices 28,29,30,31 from page 104-105 from the book "A first course in
If someone can help me, i would really appreciate it
Transcribed Image Text:28. Old Man River ... In Figure 3.2.8(a) suppose that the
y-axis and the dashed vertical line x = 1 represent, re-
spectively, the straight west and east bheaches of a river
that is 1 mile wide. The river flows northward with a
velocity v, where Iv,l = v, mi/h is a constant. A man
enters the current at the point (1, 0) on the east shore and
swims in a direction and rate relative to the river given by
the vector v, where the speed Iv,l = v, mi/h is a constant.
The man wants to reach the west beach exactly at (0, 0)
and so swims in such a manner that keeps his velocity
vector v, always directed toward the point (0, 0). Use
Figure 3.2.8(b) as an aid in showing that a mathematical
model for the path of the swimmer in the river is
dy _ vy – v,V² + y?
dx
[Hint: The velocity v of the swimmer along the path or
curve shown in Figure 3.2.8 is the resultant v = v, + v,.
Resolve v, and v, into components in the x- and
swimmer
west
east
beach
beach
current
(0, 0)
(1, 0) *
(a)
(x(1), y(t))
y()
(0, 0)
x(1)
(1, 0) *
(b)
![y-directions. If x = x(1), y = y(t) are parametric equa-
tions of the swimmer's path, then v = (dx/dt, dy/dt).]
29. (a) Solve the DE in Problem 28 subject to y(1) = 0. For
convenience let k = v,/v,-
(b) Determine the values of v, for which the swimmer
will reach the point (0, 0) by examining_ lim y(x) in
the cases k = 1, k>1, and 0<k< 1.
30. Old Man River Keeps Moving . Suppose the man in
Problem 28 again enters the current at (1, 0) but this
time decides to swim so that his velocity vector v, is
always directed toward the west beach. Assume that the
speed Iv,l = v, mi/h is a constant. Show that a mathe-
matical model for the path of the swimmer in the river
is now
dy
dx
31. The current speed v, of a straight river such as that in
Problem 28 is usually not a constant. Rather, an approxi-
mation to the current speed (measured in miles per hour)
could be a function such as v,(x) = 30x(1 – x),
0 x 1, whose values are small at the shores (in this
case, v,(0) = 0 and v,(1) = 0) and largest in the middle of
the river. Solve the DE in Problem 30 subject to y(1) = 0,
where v, = 2 mi/h and v,(x) is as given. When the swim-
mer makes it across the river, how far will he have to
walk along the beach to reach the point (0, 0)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fa0792f-28c1-40f7-9d8e-b38407416533%2Fa2b6ec82-e7f6-4f09-b398-ad358ef8996d%2Fgio49p8_processed.png&w=3840&q=75)
Transcribed Image Text:y-directions. If x = x(1), y = y(t) are parametric equa-
tions of the swimmer's path, then v = (dx/dt, dy/dt).]
29. (a) Solve the DE in Problem 28 subject to y(1) = 0. For
convenience let k = v,/v,-
(b) Determine the values of v, for which the swimmer
will reach the point (0, 0) by examining_ lim y(x) in
the cases k = 1, k>1, and 0<k< 1.
30. Old Man River Keeps Moving . Suppose the man in
Problem 28 again enters the current at (1, 0) but this
time decides to swim so that his velocity vector v, is
always directed toward the west beach. Assume that the
speed Iv,l = v, mi/h is a constant. Show that a mathe-
matical model for the path of the swimmer in the river
is now
dy
dx
31. The current speed v, of a straight river such as that in
Problem 28 is usually not a constant. Rather, an approxi-
mation to the current speed (measured in miles per hour)
could be a function such as v,(x) = 30x(1 – x),
0 x 1, whose values are small at the shores (in this
case, v,(0) = 0 and v,(1) = 0) and largest in the middle of
the river. Solve the DE in Problem 30 subject to y(1) = 0,
where v, = 2 mi/h and v,(x) is as given. When the swim-
mer makes it across the river, how far will he have to
walk along the beach to reach the point (0, 0)?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
