math 2280 - assignment 1zwick/classes/fall2013_2280/assignments/as… · section 1.1 - differential...
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Math 2280 - Assignment 1
Dylan Zwick
Fall 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
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Section 1.1 - Differential Equations and Mathe-
matical Models
1.1.1 Verify by substitution that the given function is a solution of thegiven differential equation. Throughout these problems, primes de-note 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 thegiven differential equation.
x2y′′ − xy′ + 2y = 0; y1 = x cos (ln x), y2 = x sin (ln x).
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1.1.15 Substitute y = erx into the given differential equation to determineall values of the constant r for which y = erx is a solution of theequation
y′′ + y′ − 2y = 0
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1.1.20 First verify that y(x) satisfies the given differential equation. Thendetermine a value of the constant C so that y(x) satisfies the giveninitial condition.
y′ = x − y; y(x) = Ce−x + x − 1, y(0) = 10
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1.1.45 Suppose a population P of rodents satisfies the differential equationdP/dt = kP 2. Initially, there are P (0) = 2 rodents, and their numberis increasing at the rate of dP/dt = 1 rodent per month when thereare P = 10 rodents. How long will it take for this population to growto a hundred rodents? To a thousand? What’s happening here?
Note - In order to solve this problem you need a result from problem43 of this section. Namely, that a differential equation of this formhas a solution of the form:
P (t) =1
(C − kt).
6
More room for Problem 1.1.45, if you need it.
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Section 1.2 - Integrals as General and Particular
Solutions
1.2.1 Find a function y = f(x) satisfying the given differential equationand the prescribed initial condition.
dy
dx= 2x + 1; y(0) = 3.
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1.2.6 Find a function y = f(x) satisfying the given differential equationand the prescribed initial condition.
dy
dx= x
√x2 + 9 y(−4) = 0.
9
1.2.11 Find the position function x(t) of a moving particle with the givenacceleration a(t), initial position x0 = x(0), and initial velocity v0 =v(0).
a(t) = 50,
v0 = 10,
x0 = 20.
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1.2.15 Find the position function x(t) of a moving particle with the givenacceleration a(t), initial position x0 = x(0), and initial velocity v0 =v(0).
a(t) = 4(t + 3)2,
v0 = −1,
x0 = 1.
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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 aspeed of 60m/s. How tall is the building?
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1.2.35 A stone is dropped from rest at an initial height h above the surfaceof the earth. Show that the speed with which it strikes the ground isv =
√2gh.
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1.2.43 Arthur Clark’s The Wind from the Sun (1963) describes Diana, a space-craft propelled by the solar wind. Its aluminized sail provides it witha constant acceleration of 0.001g = 0.0098m/s2. Suppose this space-craft starts from rest at time t = 0 and simultaneously fires a projec-tile (straight ahead in the same direction) that travels at one-tenth ofthe speed c = 3×108m/s of light. How long will it take the spacecraftto catch up with the projectile, and how far will it have traveled bythen?
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1.3.1 and 1.3.6 See Below
FIGURE 1.3.16.
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 in each slope field.
( —v — sinxclx
s._
13)
ixis
ofalue
(14)
f the
dy4. —=x—v
clx
3
7
\ \ \ \ \I\
\ \ \/1
/ /1/
/ \\\\‘\\\\\ III /1/1/
I Il//I 1.-—-—————--—....—._._ki
2
0
—j
/11.1.1 / / I I
/7 / IIII/Ill’
I I I \\\\/ I I I I \
/ I I III I S S s’..
———‘//1//
—1
—2
\ \ —
\ /
7/,
7/,,
— ////I /I/I//I,,
7/7/f//I //1/fl /1(11,/If/I’/ fl/I//I/ / ,b / ,•, // I/IlI / I 1
7, /1/7/7
/ /1/1/1/—, / I / I / / / I
‘‘‘‘1/I,,/7/11 1/1/f/I I/Il/f
/ I / /•/ / III I//I /I I
l/ I
I// /
/ 1,/Ill /1
_/ I I ,•/ / I.‘‘‘I,,,,’
/ /‘I
/ / /I’
/ /
—2 —1 0 I 2
FIGURE 1.3.15.
—2 —i U 1 2
FIGURE 1.3.18.
5. = y — x + 1clx
rigin
ider an Fig.
2. - = x ± vclx
3
.3
/ /,I,, /1/
I/I, /!1Il5IIl
---—‘‘‘.‘ -1d’ ‘‘ I/ 7/1/7/
S S..———I — /1//Il/Il\ 7/I/I/IIS 7//llIII IIll ‘.S’.-— ////IIII
,::---—
i/I\\\SS.. ——.,I//1
I/l\\\S /1/f
.‘.I I / I I‘‘ ..-j- — — — /
7e1111//Ill S—-———7///
I I / / II I/S ‘—‘—————/ /
/IIlIlII\ I\.—.-—7/
•I I / / / 7 I I I \ S S
III / / I / II \I\\.————
/ I / /5 I / I I S S S — — —
( 3
I/
/ / /1/
/ / - - / //1
/ //1/ ,/
lssIIIili /1/fl/I —
ill/Il//I //J///1111/Ill? I/I/IlI/Ill/Il I//I 7———
I / / / /1//I /1//•///9//•II/I/Il//I / SS
‘/
-3
7
5.-, C
—1
—2
—7
—3—2 —1
U
7
S\S\
II.•—sS\ I III
/ /SI\l / Ill
•S \ I I/ II/Ill Ill’I / I / / / 51 / /
dy3. —=y—sinx
dx
3
—3 —2 —1 0
FIGURE 1.3.19.
= x - v + I
2
I I/1
/ / /I,
/ / /,/
/ / //7
/ /Ill/Il I/I 7/I//I/Il
/ 1,1 / 1W / .7 / //‘ / 1_
111/11/1/ I, 7/7//Il
I//I/Il/I /1/
I / / / / / .5 / / —
‘
the lefthus they many
iranteeS
throughsolutionnstanCe,thole x
3
7
5.-. 0
S \\‘.-..S.. S S—2 I S 5. I I\ISS5\\\\
-3—3 —2 —l 3
‘S S I I I I ‘s S 5,
SII\\\\\S
Ill/Il’’III Il/Il’
I /•/ / \, / I
I I / / I’ / III / I, I I I / I
///1/ \\//\\
SS11J
I/Il Il/IS /
,I.IlIö\I’IS “.._——_-7/
/ / / / /1155 I/I
II Il/Il 7//I
III\\I Sfl.’———— 7/I/I\•5\5\5..,____ —, /11/1/Ills S’.”. 7/I//I IIIIss_.___ 7/Il//fl,,.,j., I ,.,.,.I, ,.
I 2 3
— / / /
II / /————‘I//I
—2 -_,p,,,.,,
7/I / / / / / /17/ /11 / /
—3 I—3 —2 —I
‘/1/1/ / II/ / / / I’I! /
/ / / / / / / // I / / III / /II / Il/Il // / / / / Ill /
•I / ,/ /
I
/ /i//Il / III
/ I ii I
x
I 2
FIGURE 1.3.17. FIGURE 1.3.20.
1.3.9 See Below
3 2_.._V_i.VVV__, /1/1,1,,,-.—--——
V
/1? / / /1 /
—. — / / / / / I., I /
/11//I//I
-
E’s.—1
—.,
S S S S S S
S S S S S S S
555555555
S S ‘•5 S S S “ S
S 5 5 5 5 \ S S
\ S S S S S S
S S S S S \ S S
H———555
——V.5
((((5 5V._V__V.. ‘555(55(\\\5V. 555555
((((55 55 ‘5\\\(\
((((((S S 555(5(5
5(5’. 5555
((((((55’. SS\
( Is’s “‘\
(((((5 \5555
‘—‘‘
—2 —i U I 2 3
FIGURE 1.3.21.
dv8. ——=.v-—v
dx
3
2
(/I_V_5I\\\ S S(lI_.—SS(( (((‘—‘/1)
/ / / — S S S S S S S — 5(/VVVS\\ 555”—— I
I I //‘—sSS S5—,/I
(5/!’— ‘_555VV._ 5/I•19._L.5
I//_ ——‘-———‘ II
(S ‘‘:‘ VV’
/5/ ‘———‘-V ///((( I/h’’ ‘‘Il//I
I ( ///// (I! Ill//I I//I/Il III Il//f! I//I/Il III (III I h//Il/I I( I /•/ / / -, I I (‘I (I/Il/I/I / /llIIIIIi(
iI I I I II
III
I(I
3 I ) 1 2
—‘5
—2 -I
FIGURE 1.3.22.
dv9. —-x—v—2
dx -
3
-5
—‘5
-2 -1 0 2
FIGURE 1.3.24.
In Problems 11 through 20, dete,,nine whether Theorem I does
or does not guarantee existence ofa solution oft/ic given initial
ma/ac problem. hf existence is guaranteed, determine whether
Jimeoremn I does or does not guarantee uniqueness of that solution.
(I V= 2.v2v vU) = —1
(/.V
d12. — =xlnv: vU) = 1
dx
13. - = v(O)
14. = (0) = 0dx
15. -=,jr-; y(2)=2
16. -=,/‘‘; v(2)=1
dv17. v——=x—1; v(0)=1
dx
£1 V18. v— = x — 1; v(1) = 0
dx
19. = In(1 + v2); y(O) = 0
20. —- = x2 — v2; v(0) = 1(LU
In Probleni.s 2 I and 22, first use the method of’ LUamnple 2
to construct a slope field for the given diffrrential equation.
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).
21. v’=rx+v, v(0)=0; v(—4)=?
//I\\(1(/5
I /—\\ 5 I S S 55—I /
‘‘-.‘.‘‘ II\\5.’/(f_.\((\\ \III\—_V_/
(//‘—S(I\ SiI/_ S/Il ((555—/Si
/N..\( S
3
2
—l
—2
( I / — — S S S(///—55
(1/III’’————
I / / /
S I I / “
II /1/,—5/ I / / “ ‘
H.V———, /5 5
/ / I/ I
_// / / II I— / /5 / / I I I
—3 —2 —l 0 1 2 3S
FIGURE 1.3.23. 22. v’==v—x, y(4)=O; v(—4)=?
1.3.11 Determine whether Theorem 1 does or does not guarantee existenceof a solution of the given initial value problem. If existence is guar-anteed, determine whether Theorem 1 does or does not guaranteeuniqueness of that solution.
dy
dx= 2x2y2 y(1) = −1.
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1.3.15 Determine whether Theorem 1 does or does not guarantee existenceof a solution of the given initial value problem. If existence is guar-anteed, determine whether Theorem 1 does or does not guaranteeuniqueness of that solution.
dy
dx=
√x − y y(2) = 2.
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1.3.21 First use the method of Example 2 from the textbook to construct aslope field for the given differential equation. Then sketch the solu-tion curve corresponding to the given initial condition. Finally, usethis solution curve to estimate the desired value of the solution y(x).
y′ = x + y, y(0) = 0; y(−4) =?
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More room for Problem 1.3.21, if you need it.
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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
satisfies the differential equation y′ = 3y2
3 for all x. Can you alsouse the “left half” of the cubic y = (x − c)3 in piecing together asolution curve of the differential equation? Sketch a variety of suchsolution curves. Is there a point (a, b) of the xy-plane such that the
initial value problem y′ = 3y2
3 , y(a) = b has either no solution or aunique solution that is defined for all x? Reconcile your answer withTheorem 1.
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More room for Problem 1.3.29, if you need it.
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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
dx+ 2xy = 0
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1.4.3 Find the general solution (implicit if necessary, explicit if convenient)to the differential equation
dy
dx= y sin x
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1.4.17 Find the general solution (implicit if necessary, explicit if conve-nient) to the differential equation
y′ = 1 + x + y + xy.
Primes denote the derivatives with respect to x. (Suggestion: Factorthe right-hand side.)
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1.4.19 Find the explicit particular solution to the initial value problem
dy
dx= yex, y(0) = 2e.
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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 theinitial value problem y′ = 2
√y, y(a) = b has (a) no solution, (b) a
unique solution, (c) infinitely many solutions.
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More room for Problem 1.4.31, if you need it.
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1.4.35 (Radiocarbon dating) Carbon extracted from an ancient skull con-tained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull?
29
1.4.53 Thousands of years ago ancestors of the Native Americans crossedthe 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 hassince split into many Indian “language families.” Assume that thenumber of these language families has been multiplied by 1.5 every6000 years. There are now 150 Native American language familiesin the western hemisphere. About when did the ancestors of today’sNative Americans arrive?
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1.4.68 The figure below shows a bead sliding down a frictionless wirefrom point P to point Q. The brachistochrone problem asks what shapethe wire should be in order to minimize the bead’s time of descentfrom P to Q. In June of 1696, John Bernoulli proposed this problemas a public challenge, with a 6-month deadline (later extended toEaster 1697 at George Leibniz’s request). Isaac Newton, then retiredfrom academic life and serving as Warden of the Mint in London, received Bernoulli’s challenge on January 29, 1697. The very next dayhe communicated his own solution - the curve of minimal descenttime is an arc of an inverted cycloid - to the Royal Society of London.For a modern derivation of this result, suppose the bead starts fromrest at the origin P and let y y(x) be the equation of the desiredcurve in a coordinate system with the y-axis pointing downward.Then a mechanical analogue of Snell’s law in optics implies that
sin ct= constant
V
where c denotes the angle of deflection (from the vertical) of the tangent line to the curve - so cot c y’(x) (why?) - and v /2jj is thebead’s velocity when it has descended a distance y vertically (fromKE = mv2 = mgy = -PE).
p
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(a) First derive from sin α/v = constant the differential equation
dy
dx=
√
2a − y
y
where a is an appropriate positive constant.
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(b) Substitute y = 2a sin2 t, dy = 4a sin t cos tdt in the above differen-tial 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 parametricequations x = a(θ − sin θ), y = a(1 − cos θ) of the cycloid that isgenerated by a point on the rim of a circular wheel of radius aas it rolls along the x-axis.
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