Math 2280 - Assignment 1
Dylan Zwick
Summer 2013
Section 1.1 - 1, 12, 15, 20, 45
Section 1.2 - 1, 6, 11, 15, 27, 35, 43
Section 1.3 - 1, 6, 9, 11, 15, 21, 29
Section 1.4 - 1, 3, 17, 19, 31, 35, 53, 68
1
Section 1.1 - Differential Equations and Mathematical Models
1.1.1 Verify by substitution that the given function is a solution of the
given differential equation. Throughout these problems, primes denote derivatives with respect to x.
y ′ = 3x2 ;
y = x3 + 7
2
1.1.12 Verify by substitution that the given function is a solution of the
given differential equation.
x2 y ′′ − xy ′ + 2y = 0;
y1 = x cos (ln x),
3
y2 = x sin (ln x).
1.1.15 Substitute y = erx into the given differential equation to determine
all values of the constant r for which y = erx is a solution of the
equation
y ′′ + y ′ − 2y = 0
4
1.1.20 First verify that y(x) satisfies the given differential equation. Then
determine a value of the constant C so that y(x) satisfies the given
initial condition.
y ′ = x − y;
y(x) = Ce−x + x − 1,
5
y(0) = 10
1.1.45 Suppose a population P of rodents satisfies the differential equation
dP/dt = kP 2 . Initially, there are P (0) = 2 rodents, and their number
is increasing at the rate of dP/dt = 1 rodent per month when there
are P = 10 rodents. How long will it take for this population to grow
to a hundred rodents? To a thousand? What’s happening here?
6
More room for Problem 1.1.45, if you need it.
7
Section 1.2 - Integrals as General and Particular
Solutions
1.2.1 Find a function y = f (x) satisfying the given differential equation
and the prescribed initial condition.
dy
= 2x + 1;
dx
8
y(0) = 3.
1.2.6 Find a function y = f (x) satisfying the given differential equation
and the prescribed initial condition.
√
dy
= x x2 + 9
dx
9
y(−4) = 0.
1.2.11 Find the position function x(t) of a moving particle with the given
acceleration a(t), initial position x0 = x(0), and initial velocity v0 =
v(0).
a(t) = 50,
v0 = 10,
x0 = 20.
10
1.2.15 Find the position function x(t) of a moving particle with the given
acceleration a(t), initial position x0 = x(0), and initial velocity v0 =
v(0).
a(t) = 4(t + 3)2 ,
v0 = −1,
x0 = 1.
11
1.2.27 A ball is thrown straight downward from the top of a tall building.
The initial speed of the ball is 10m/s. It strikes the ground with a
speed of 60m/s. How tall is the building?
12
1.2.35 A stone is dropped from rest at an initial height h above the surface
of the
√ earth. Show that the speed with which it strikes the ground is
v = 2gh.
13
1.2.43 Arthur Clark’s The Wind from the Sun (1963) describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with
a constant acceleration of 0.001g = 0.0098m/s2. Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of
the speed c = 3×108m/s of light. How long will it take the spacecraft
to catch up with the projectile, and how far will it have traveled by
then?
14
1.3.1 and 1.3.6 See Below
In Pivblenis I thmugh 10, ne hai’e provided the slope field of
the indicated difft’i-ential equation, together with one or inon’
solution curves. Sketch likel’ solution curves through the ad
ditional points marked each slope field.
13)
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FIGURE 1.3.21.
In Problems 11 through 20, dete,,nine whether Theorem I does
of a solution oft/ic given initial
or does not guarantee
is guaranteed, determine whether
ma/ac problem. hf
that so
Jimeoremn I does or does not guarantee uniqueness
existence
of
existence
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16.
v(2)=1
FIGURE 1.3.22.
dv
9. —-x—v—2
dx
-
3
2
5
5
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v(0)=1
£1 V
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dx
v(1)
=
x
—
1;
=
0
19.
=
In(1 + v
);
2
=
2
x
y(O)
=
0
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3
LUamnple 2
In Probleni.s 2 I and 22, first use the method
diffrrential
equation.
given
the
for
field
slope
a
construct
to
Then sketch the solution cane corresponding to the given ini
tial condition. Finally, use this solution cane to estimate the
desired ia/ac oft/ic solution v(x).
S
21.
FIGURE 1.3.23.
v’=rx+v,
22. v’==v—x,
v(0)=0;
y(4)=O;
v(—4)=?
v(—4)=?
1.3.11 Determine whether Theorem 1 does or does not guarantee existence
of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee
uniqueness of that solution.
dy
= 2x2 y 2
dx
17
y(1) = −1.
1.3.15 Determine whether Theorem 1 does or does not guarantee existence
of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee
uniqueness of that solution.
dy √
= x−y
dx
18
y(2) = 2.
1.3.21 First use the method of Example 2 from the textbook to construct a
slope field for the given differential equation. Then sketch the solution curve corresponding to the given initial condition. Finally, use
this solution curve to estimate the desired value of the solution y(x).
y ′ = x + y,
y(0) = 0;
19
y(−4) =?
More room for Problem 1.3.21, if you need it.
20
1.3.29 Verify that if c is a constant, then the function defined piecewise by
y(x) =
0
x ≤ c,
(x − c)3 x > c
2
satisfies the differential equation y ′ = 3y 3 for all x. Can you also
use the “left half” of the cubic y = (x − c)3 in piecing together a
solution curve of the differential equation? Sketch a variety of such
solution curves. Is there a point (a, b) of the xy-plane such that the
2
initial value problem y ′ = 3y 3 , y(a) = b has either no solution or a
unique solution that is defined for all x? Reconcile your answer with
Theorem 1.
21
More room for Problem 1.3.29, if you need it.
22
Section 1.4 - Separable Equations and Applications
1.4.1 Find the general solution (implicit if necessary, explicit if convenient)
to the differential equation
dy
+ 2xy = 0
dx
23
1.4.3 Find the general solution (implicit if necessary, explicit if convenient)
to the differential equation
dy
= y sin x
dx
24
1.4.17 Find the general solution (implicit if necessary, explicit if convenient) to the differential equation
y ′ = 1 + x + y + xy.
Primes denote the derivatives with respect to x. (Suggestion: Factor
the right-hand side.)
25
1.4.19 Find the explicit particular solution to the initial value problem
dy
= yex ,
dx
y(0) = 2e.
26
1.4.31 Discuss the difference between the differential equations (dy/dx)2 =
√
4y and dy/dx = 2 y. Do they have the same solution curves? Why
or why not? Determine the points (a, b) in the plane for which the
√
initial value problem y ′ = 2 y, y(a) = b has (a) no solution, (b) a
unique solution, (c) infinitely many solutions.
27
More room for Problem 1.4.31, if you need it.
28
1.4.35 (Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much 14 C as carbon extracted from presentday bone. How old is the skull?
29
1.4.53 Thousands of years ago ancestors of the Native Americans crossed
the Bering Strait from Asia and entered the western hemisphere.
Since then, they have fanned out across North and South America.
The single language that the original Native Americans spoke has
since split into many Indian “language families.” Assume that the
number of these language families has been multiplied by 1.5 every
6000 years. There are now 150 Native American language families
in the western hemisphere. About when did the ancestors of today’s
Native Americans arrive?
30
1.4.68 The figure below shows a bead sliding down a frictionless wire
from point P to point Q. The brachistochrone problem asks what shape
the wire should be in order to minimize the bead’s time of descent
from P to Q. In June of 1696, John Bernoulli proposed this problem
as a public challenge, with a 6-month deadline (later extended to
Easter 1697 at George Leibniz’s request). Isaac Newton, then retired
from academic life and serving as Warden of the Mint in London, received Bernoulli’s challenge on January 29, 1697. The very next day
he communicated his own solution - the curve of minimal descent
time is an arc of an inverted cycloid - to the Royal Society of London.
For a modern derivation of this result, suppose the bead starts from
rest at the origin P and let y = y(x) be the equation of the desired
curve in a coordinate system with the y-axis pointing downward.
Then a mechanical analogue of Snell’s law in optics implies that
sin α
= constant
v
where α denotes the angle of deflection (from the vertical)√of the tangent line to the curve - so cot α = y ′ (x) (why?) - and v = 2gy is the
bead’s velocity when it has descended a distance y vertically (from
KE = 12 mv 2 = mgy = −P E).
31
(a) First derive from sin α/v = constant the differential equation
dy
=
dx
r
2a − y
y
where a is an appropriate positive constant.
32
(b) Substitute y = 2a sin2 t, dy = 4a sin t cos tdt in the above differential equation to derive the solution
x = a(2t − sin 2t),
y = a(1 − cos 2t)
for which t = y = 0 when x = 0. Finally, the substitution of θ =
2a in the equations for x and y yields the standard parametric
equations x = a(θ − sin θ), y = a(1 − cos θ) of the cycloid that is
generated by a point on the rim of a circular wheel of radius a
as it rolls along the x-axis.
33