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Assignment #2 (16015237)

1. -Question Details ZillDiffEQ9 2.1.001. [3876604]

Reproduce the given computer-generated direction field. Then sketch an approximate solution curve that passes through eachof the indicated points.

(a)

= x − ydydx

2 2

y(−2) = 1



(b) y(3) = 0



(c) y(0) = 2



(d) y(0) = 0





(a)

= 1 − xydydx

y(0) = 0

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(b) y(−1) = 0



(c) y(2) = 2



(d) y(0) = −3





= (sin(x))cos(y)dydx




(a) y(0) = 1



(b)

(c)

y(1) = 0

y(2) = 2



(d) y(0) = − 32




The given figure represents the graph of f(y). Sketch a direction field over an appropriate grid for dy/dx = f(y).





The given figure represents the graph of f(x). Sketch a direction field over an appropriate grid for dy/dx = f(x).




Consider the following autonomous first-order differential equation.

Find the critical points and phase portrait of the given differential equation.

Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability.Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.)

= y − 4ydydx

2

ℹ ℹ ℹ ℹ




asymptotically stable

unstable

semi-stable

Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

ℹ ℹ

ℹ ℹ





Consider the following autonomous first-order differential equation.

Find the critical points and phase portrait of the given differential equation.

Classify each critical point as asymptotically stable, unstable, or semi-stable. (List the critical points according to their stability.Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.)

asymptotically stable

unstable

semi-stable

Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

= y (16 − y )dydx

2 2




The number of supermarkets throughout the country that are using a computerized checkout system is described by theinitial-value problem

N(t)

= N(1 − 0.0008N), N(0) = 1.dNdt




(a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the newprocedure over a long period of time.

supermarkets

By hand, sketch a solution curve of the given initial-value problem.

(b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a).

How many supermarkets are expected to adopt the new technology when (Round your answer to the nearestinteger.)

supermarkets

ℹ ℹ

ℹ ℹ

N(t) =

t = 5?





Investigate the following harvesting model both qualitatively and analytically.

If a constant number h of fish are harvested from a fishery per unit time, then a model for the population of the fishery attime t is given by

where are positive constants. Suppose

Determine whether the population becomes extinct in finite time.

If so, find that time. (If not, enter NONE.)

P(t)

= P(a − bP) − h, P(0) = P ,dPdt 0

a, b, h, and P0 a = 9, b = 1, and h = .814

The population becomes extinct in finite time for all values of

The population becomes extinct in finite time if


The population does not become extinct in finite time.


P .0

P < .092

P > .092

P = .092

t =


(a) Suppose in the Gompertz differential equation,

(see (7) in Section 3.2). Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representativesolution curves corresponding to the cases

a = b = 1

= P(a − b ln(P))dPdt

P > e and 0 < P < e.0 0




(b) Suppose in the Gompertz differential equation. Use a new phase portrait to sketch representative solution

curves corresponding to the cases and

(c) Find an explicit solution of the Gompertz differential equation subject to

a = 1, b = −1

P > e0−1 0 < P < e .0

−1

P(0) = P .0

P(t) =





Find the general solution of the given differential equation.

Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enteryour answer using interval notation.)

Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separatedlist; if there are none, enter NONE.)

y' + 7x y = x6 6

y(x) =



Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enteryour answer using interval notation.)


x y' + xy = 22

y(x) =







Give the largest interval over which the general solution is defined. (Think about the implications of any singular points.)


cos(x) + (sin(x))y = 1dydx

y(x) =

0, π2

(−∞, ∞)

(−1, 1)

− , π2π2

(0, π)


Solve the given initial-value problem.

Give the largest interval I over which the solution is defined. (Enter your answer using interval notation.)

xy' + y = e , y(1) = 8x

y(x) =

I =





Assignment Details


Solve the given initial-value problem.

Give the largest interval I over which the solution is defined. (Enter your answer using interval notation.)

x + y = 6x + 1, y(1) = 6dydx

y(x) =

I =


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