Homework-4-f11-solution

Homework-4-f11-solution - AMS 361: Applied Calculus IV (DE...

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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 1-separable, 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 1-separable, 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: Re-group 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 anti-derivative 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 Medium-1 is , in Medium-2 is , and Medium-3 is . Compute the bullet’s speed to enter Medium-1 such that the bullet will pass through Medium-1 and then Medium-2 and then Medium-3, and magically stop precisely at the edge of Medium-3. You may neglect gravity. Solution: Assume the following speeds: and are entering and exiting Medium-1, respectively. and are entering and exiting Medium-2, respectively. and are entering and exiting Medium-3, respectively. For Medium-1: the equation of motion is or Integrating both sides, we get: Thus, the solution is given by: For Medium-2: treat it using the same strategy for Medium-1. or Integrating both sides, we get: Thus, the solution is given by: For Medium-3, we know easily (from Medium-1) 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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