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Unformatted text preview: AMS 361: Applied Calculus IV (DE & BVP)
Homework 4
Assignment Date: Wednesday (10/05/2011) Collection Date: Thursday (10/13/2011) Grade: Each problem is worth 10 points Problem 4.1: The following differential equation is of two different types considered in
Chapter 1separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
solutions for the given equation in two different ways, and then reconcile your results:
Solution:
Method 1: Separation of variables ∴
Method 2: Integration factor
Find the integration factor,
Multiply the integration factor and take an integral, ∴ 1 Problem 4.2: The following differential equation is of two different types considered in
Chapter 1separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general
solutions for the given equation in two different ways, and then reconcile your results:
Solution:
Method 1: Exact DE method:
Two parameter functions are listed below:
Calculate the difference:
This is Exact D.E. . Expand the original equations and track back for the solution:
Regroup the terms as
Or
Or
Thus, the solution is
Of course, one can also go through the extensive Exact DE method to get the solution.
The above approach can be handy for simple problems.
Method 2: Substitution:
Introducing a new function by We get
,
whose solution is
Finally, the solution is: 2 Problem 4.3: Consider a prolific animal whose birth and death rates, and , are each
proportional to its population with
and
.(a) Compute the population
as a function of time and parameters and initial condition given; (b) Find the time for
doomsdday; (c) Suppose that
and that there are 4027 animals after 12 time
units (days or months or years), when is the doomsday? (d) If
, compute the
population limit when time approaches infinity?
Solution:
With , and , population model is given by: Take the antiderivative and gives:
(a) Coupling with Initial Condition, where
solution as: , and locate the (b) Calculate the doomsday formula:
The denominator is zero, which satisfies the definition of doomsday. So, time is
given below: (c) Calculate the doomsday time:
First, calculate the difference between the birth rate and death rate: Second, calculate the time: (d) Calculate the population:
Due to , we have . So, As time goes to infinity, the population goes to zero. 3 Problem 4.4: Consider shooting a bullet of mass to three connected (and fixed) media
each of equal length . The resistance in Medium1 is
, in Medium2 is
, and
Medium3 is
. Compute the bullet’s speed to enter Medium1 such that the bullet will
pass through Medium1 and then Medium2 and then Medium3, and magically stop
precisely at the edge of Medium3. You may neglect gravity.
Solution:
Assume the following speeds:
and
are entering and exiting Medium1, respectively.
and
are entering and exiting Medium2, respectively.
and
are entering and exiting Medium3, respectively.
For Medium1: the equation of motion is
or
Integrating both sides, we get: Thus, the solution is given by: For Medium2: treat it using the same strategy for Medium1.
or
Integrating both sides, we get: Thus, the solution is given by: For Medium3, we know easily (from Medium1)
Force and we have:
or Trace back and find the final solution:
∴ 4 Problem 4.5: Suppose that the fish population
in a lake is attacked by a disease
(such as human being who eats them) at time
, with the result that the fish cease to
reproduce (so that the birth rate is
) and the death rate (death per week per fish) is
thereafter proportional to
. If there were initially 4000 fish in the lake and 2011
were left after 11 weeks, how long did it take all the fish in the lake to die? Can you
change the “2011” to a different number such that the fish count never changes with time?
To what number if so?
Solution:
(a) The population modeling is given by: The solution is followed by:
When we couple with the initial condition, we get:
When we plug in the population value of 2011 with respect to time 11, the unknown
coefficient is: So, the time for all the fish in the lake to die(let
(b) If the population does not change with time,
2011. ):
. So we need 4000 rather than 5 ...
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