Math 2280 - Assignment 2

Dylan Zwick

Spring 2013

Section 1.4 - 1, 3, 17, 19, 31, 35, 53, 68

Section 1.5 - 1, 15, 21, 29, 38, 42

Section 1.6 - 1, 3, 13, 16, 22, 26, 31, 36, 56

1

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 = 0dx

—x

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2

1.4.3 Find the general solution (implicit if necessary, explicit if convenient)to the differential equation

dy— y Sill Xdx

3

1.4.17 Find the general solution (implicit if necessary, explicit if convenient) to the differential equation

= 1 + x + y + y.

Primes denote the derivatives with respect to x. (Suggestion: Factorthe right-hand side.)

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(±( t*y)

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4

1.4.19 Find the explicit particular solution to the initial value problem

dyy(O)2e.

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5

1.4.31 Discuss the difference between the differential equations (dy/dx)2 =

4y and dy/di = 2v• Do they have the same solution curves? Whyor why not? Determine the points (a, b) in the plane for which theinitial value problem y’ = 2, y(a) 5 has (a) no solution, (b) aunique solution, (c) infinitely many solutions.

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6

1.4.35 (Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much ‘4C as carbon extracted from present-day bone. How old is the skull?

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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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8

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 a= constant

V

where a denotes the angle o1 deflection (from the vertical) of the tangent line to the curve - so y’(x) (why?) - and v = is thebead’s velocity when it has descended a distance y vertically (fromKE = mv2 = rngy -PE).

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(b) Substitute y = 2a sin2 t. dy = 4a sin t cos tdt in the above differential equation to derive the solution

x=a(2t—sin2t), yrra(1—cos2t)

for which t y = 0 when x 0. Finally, the substitution of 0 =

2a in the equations for x and y yields the standard parametricequations x = a(0 — sin6),y a(1 — cosO) 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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11

Section 1.5 - Linear First-Order Equations

1.5.1 Find the solution to the initial value problem

y’+y2 y(O)=O

x

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y()\/(ô)

12

1.5.15 Find the solution to the initial value problem

t

y’ + 2xy = y(O) = —2.

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1.5.29 Express the general solution of dy/dx 1 + 2xy in terms of the errorfunction

erf(x) — f e2dt.

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15

1.5.38 Consider the cascade of two tanks shown below with V1 = 100 (gal)and V2 = 200 (gal) the volumes of brine in the two tanks. Each tankalso initially contains 50 lbs of salt. The three flow rates indicated inthe figure are each 5 gal/mm, with pure water flowing into tank 1.

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(a) 1. at time t.

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16

(b) Suppose that y(t) is the amount of salt in tank 2 at time t. Showfirst that

dy 5x 5y-Th-o.

and then solve for y(t), using the function x(t) found in part (a).

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(c) Finally, find the maximum amount of salt ever in tank 2.

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1.5.42 Suppose that a falling hailstone with density 6 = 1 starts from restwith negligible radius r = 0. Thereafter its radius is r = kt (k is a constant) as it grows by accreation during its fall. Use Newton’s second law - according to which the net force F acting on a possibly variable mass rn equals the time rate of change dp/dt of its momentump = mu - to set up and solve the initial value problem

(mv) = mg, v(0) = 0,

where m is the variable mass of the hailstone, v = cly/dt is its velocity.and the positive y-axis points downward. Then show that dv/dt =

g/4. Thus the hailstone falls as though it were under one-fourth theinfluence of gravity.

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1.6.13 Find the general solution of the differential equation

+ + y2

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(v+ V1/L ) = iv?

v+q 6(4

y

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1.6.26 Find the general solution of the differential equation

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3

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3.

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e

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3

24

1.6.31 Verify that the differential equation

(2x + 3y)dx + (3x + 2y)dy 0

is exact; then solve it.

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1.6.56 Suppose that n 0 and n 1. Show that the substitutuion v y1_fl

transforms the Bernoulli equation

1 + P(x)y = Q(x)ydx

into the linear equation

— + (1- n)P(x)v(x) = (1- n)Q(x).dx

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