Homework-4-f10_Sol

Homework-4-f10_Sol - AMS 361: Applied Calculus IV (DE...

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Unformatted text preview: AMS 361: Applied Calculus IV (DE & BVP) Homework 4 Assignment Date: Collection Date: Grade: Thursday (10/07/2010) Thursday (10/14/2010) Each problem is worth 10 points Problem 4.1 The following differential equation is of two different types considered in Chapter 1-separable, linear, homogenous, Bernoulli, exact, etc. Hence, derive general solutions for the given equation in two different ways, and then reconcile your results Method 1) Separation of variables ( ) ∫ ( ) ( ∫ ( ) ) Method 2) First order () () ( ( ) ) ( () ∫ ∫ () ( ) ( ) ( ( ) ) ) 1 Problem 4.2 The following differential equation is of two different types considered in Chapter 1-separable, linear, homogenous, Bernoulli, exact, etc. Hence, derive general solutions for the given equation in two different ways, and then reconcile your results ( Exact equation method: ( ( )) ( ) ( )) () () Since () Exact equation. () () (a) (b) From () () () () () () () () () () ( Substitution method: Introduce ( and we can get ) ) 2 ( Thus, and is the solution. ) Problem 4.3 Consider a prolific breed of rabbits whose birth and death rates, are each ( ) (a) Compute the rabbit proportional to the rabbit population with and population as a function of time and parameters and initial condition given; (b) Find the time for doomsday; (c) Suppose that and that there are 15 rabbits after 12 months, when is the doomsday? (d) If , compute the population limit when time approaches infinity? Part A ( ( ) ) ∫ ∫ ( () ) ( ) Part B 3 Part C Find doomsday for () ( ) ( )( ) () Part D As Problem 4.4 Consider shooting a bullet of mass to a medium whose resistance is proportional to the speed, to a second medium whose resistance is proportional to , and a third medium whose resistance is proportional to . Compute in all three cases the farthest the bullet can travel. Part A ( ∫ [ ) ∫ ] [ ] 4 Part B ( ∫ ) ∫ [ [ Part C ( ) ∫ () ∫ () 5 Problem 4.5: A man with a parachute jumps out of a “frozen-in-sky” helicopter at height . During the fall, the man's drag coefficient is (with parachute closed) and (with parachute open) and air resistance is proportional to velocity. The total weight of the man and his parachute is Take the initial velocity to be zero. Gravitational constant is Find the best time for the man to open his parachute after he leaves the helicopter for the quickest fall and yet “soft” landing at touchdown speed . Compute the total falling time. Before opening the parachute ∫ () () () ( ∫ ) () ∫ () () () ( ( ) ) After opening the parachute ∫ () (( ∫ () ( () ) ∫ () ) ) ( ) 6 () ( ) () Solve Solve using the formula ( ) ( ) using Problem 4.6 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 (deaths per week per fish) is thereafter proportional to . If there were initially 1000 fish in the lake and 88 were left after 18 weeks, how long did it take all the fish in the lake to die? Can you change the “88” to a different number such that the fish count never changes with time? To what number if so? ( ) ( √ ∫ ) √ ∫ √ √ () √ () ( () ) ( ( √ ) √ ) () If the population does not change with time, k is 0. Therefore ( ) ( ) 7 ...
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