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Diff EQ HW 5
0. Read Chapters 3.2 and 4.3 of the Textbook.
1. The general solution to each 2nd order linear differential equation is given. Find the
constants C1 and C2 which satisfy the given initial data.
(a) y ′′ − 9y = 0, y(0) = 2, y ′ (0) = −1, y = C1 e3x + c2 e−3x
(b) y ′′ + 11y ′ + 24y = 0, y(0) = 0, y ′ (0) = −7, y = C1 e−8x + c2 e−3x
2. Use the given characteristic (factor) to solve each of the Higher order order linear
homogeneous differential equation for the general solution. (Assuming the form y =
C1 er1 x + C2 er2 x . . .)
(a) y ′′′ − 3y ′′ − y ′ + 3y = 0, r3 − 3r2 − r + 3 = 0
(b) y (4) − 3y ′′′ + 3y ′′ − y ′ = 0, r4 − 3r3 + 3r2 − r = 0
(c) y (4) − 4y ′′′ + 8y ′ + 4y = 0, (r2 − 2r + 2)2 = 0
3. The number N (t) of trucks of a certain model that were sold within t months satisfies
the following logistic differential equation:
3N
dN
= N 90, 000 −
.
dt
20, 000
There were initially 20,000,000 trucks sold. What is the carrying capacity of trucks
sold?
4. The population of hawks in a forest after t years satisfies the logistic differential equation
P
dP
= 3P 1 −
,
dt
2500
where the initial population is 1000 mice. What is the population when it’s growing
the fastest?
5. Doomsday Equation: Suppose the differential Equation
dP
= kP 1.01 , P (0) = 10 ,
dt
is a mathematical model for a population of small animals, where t is measured in
months.
(a) Solve the differential equation subject to the fact that the animal population has
doubled in 5 months.
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(b) This differential Equation is called a doomsday equation because the population
P (t) exhibits unbounded growth over a finite interval (0, T ). That is, there is
some time T such that P (T ) → ∞ as t → T − . Find T .
6. The number N (t) of people in a community who are exposed to a particular propaganda
is modeled by the logistic equation. Initially N (0) = 500, and it is observed that
N (1) = 1000. Solve for N (t) if it is predicted that the limiting number of people who
will see the propaganda is 50, 000.
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