Page 60 - Advanced engineering mathematics

40 CHAPTER 1 First-Order Differential Equations

11. In a constant electromotive force RL circuit, we find 19. A bug is located at each corner of a square table of
that the current is given by side length a. At a given time, the bugs begin moving
E at constant speed v with each pursuing the neighbor to
i(t) = (1 − e −Rt/L ) + i(0)e −Rt/L . the right.
R
Let i(0) = 0. (a) Determine the curve of pursuit of each bug. Hint:
Use polar coordinates with the origin at the center
(a) Show that the current increases with time.
of the table and the polar axis containing one of
(b) Find a time t 0 at which the current is 63 percent the corners. When a bug is at ( f (θ),θ), its target
of E/R. This time is called the inductive time is at ( f (θ,θ + π/2)). Use the chain rule to write
constant of the circuit.
dy dy/dθ
(c) Does the inductive time constant depend on i(0)? =
dx dx/dθ
If so, in what way?
where
12. Recall that the charge q(t) in an RC circuit satisfies
the linear differential equation y(θ) = f (θ)sin(θ) and x(θ) = f (θ)cos(θ).
1 1
(b) Determine the distance traveled by each bug.
q + q = E(t).
RC R
(c) Does any bug actually reach its quarry?
(a) Solve for the charge in the case that E(t) = 20. A bug steps onto the edge of a disk of radius a that
E, which is constant. Evaluate the constant of is spinning at a constant angular speed of ω.The
integration in this solution process by using the bug moves toward the center of the disk at constant
condition q(0) = 0.
speed v.
(b) Determine lim t→∞ q(t), and show that this limit is
independent of q 0 . (a) Derive a differential equation for the path of the
bug using polar coordinates.
(c) Determine at what time q(t) is within 1 percent of
its steady-state value (the limiting value requested (b) How many revolutions will the disk make before
the bug reaches the corner? (The solution will be
in part (b)).
in terms of the angular speed and radius of the
In each of Problems 13 through 17, find the family of disk).
orthogonal trajectories of the given family of curves. If
(c) Referring to part (b), what is the total distance the
software is available, graph some curves of both families.
bug will travel, taking into account the motion of
2
13. 2x − 3y = k the disk?
14. x + 2y = k 21. A 24 foot chain weighing ρ pounds per foot is
2
15. y = kx + 1 stretched out on a very tall, frictionless table with 6
feet hanging off the edge. If the chain is released from
2
2
16. x + 2y = k
rest, determine the time it takes for the end of the chain
17. y = e kx to fall off the table and also the velocity of the chain
at this instant.
18. A man stands at the junction of two perpendicular
roads, and his dog is watching him from one of the 22. Suppose the chain in Problem 21 is placed on a table
roads at a distance A feet away. At some time, the that is only 4 feet high, so that the chain accumulates
man starts to walk with constant speed v along the on the floor as it slides off the table. Two feet of chain
other road, and at the same time, the dog begins to are already piled up on the floor at the time that the
run toward the man with a speed of 2v. Determine the rest of the chain is released. Determine the velocity of
path the dog will take, assuming that it always moves the moving end of the chain at the instant it leaves the
so that it is facing the man. Also determine when the table top. Hint: Newton’s law applies to the center of
dog will eventually catch the man. mass of the moving system.

1.6 Existence and Uniqueness Questions

There are initial value problems having no solution. One example is
√

y = 2 y; y(0) =−1.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:9 THM/NEIL Page-40 27410_01_ch01_p01-42