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are important means of getting a good score on this exam. Neatness and organization are also
important. Problem Points Problem Points Problem 1 (30 points) 7 (10 points) 13 (20 points) 2 (20 points) 8 (10 points) 14 (20 points) 3 (20 points) 9 (10 points) 15 (20 points) 4 (30 points) 10 (10 points) 16 (10 points) 5 (10 points) 11 (30 points) 17 (10 points) 6 (10 points) 12 (20 points) 18 (10 points) Points QUIZ 1
1) Find the general solution to the given Ordinary Differential Equations
a) ty ' y=e 2t b) y ' −t 2 y 2=0 c) dy −4x−5y
=
dx 5x6 y 2) A tank originally contains 30 gallons of pure water. Then, water containing 2 grams of salt per
gallon is poured into the tank at a rate of 3 gallons per minute. Water leaves the tank at the same rate.
a) Write out an IVP problem which when solved will tell you how many grams of salt are in the tank at
any time t. b) Find the amount of salt in the tank at any time t. dy
= f y where f y =sin ∣ y∣ on −2.5 y2.5
dt
a) Sketch the graph of f(y) versus y, Draw the Phase Line (p), and sketch several graphs of solutions in
the ty plane (at least 1 for each fundamentally different solution). 3) Let p y f(y)          t     y     b) Find the equilibrium solutions and determine whether they are stable, unstable, or semistable. QUIZ 2
4) Find the General Solution of the given Homogeneous Differential Equations.
a) y ' ' −20 y ' 100y=0 b) y ' '−5 y ' −50y=0 c) y ' '10y '125y=0 t 5) Verify that e e
y 1 t=
2 t −t and 6) Solve the Initial Value Problem e −e
y 2 t =
2 L [ y ]=0, −t are linearly independent on any open interval I. y 0=0, y ' 0=0 . Justify your answer. 7) Given t 2 y ' ' 3t y ' y=0 has a solution
find the general solution. y 1 t= 1
, use the Reduction of Order technique to
t 8) Use the Method of Undetermined Coefficients to find the general solution to the following nonhomogeneous equations.
−10t
y ' '−20 y ' 100 y=400 e QUIZ 3
9) Find L{ e−t et
}
2 n!
} = s−a n1 , use the Laplace Transform to solve for y.
y ' '4 y ' 4y=0
y 0=0 , y ' 0=−2 10) Given L{ n t e at 11) Consider two interconnected tanks which initially contain pure water, namely Tank 1 which has 30
gallons pure water and Tank 2 which has 20 gallons pure water. A salt mixture of 2 grams per gallon is
poured into Tank 1 at a rate of 3 gallons per minute, and Tank 1 is pumping water into Tank 2 at a rate
of 5 gallons per minute. A salt mixture of 1 gram per gallon is being pumped into Tank 2 at a rate of 1
gallon per minute, and Tank 2 is pumping water into Tank 1 at a rate of 2 gallons per minute, and water
is exiting Tank 2 into the local river at a rate of 4 gallons per minute.
a) Draw a diagram to show what is happening in this waterworks system. b) Create a system of differential equations to determine that amount of salt Q1 t and Q 2 t in
Tank 1 and Tank 2. c) Transform the System of ODEs into a second order ODE by substituting the first equation into the
second equation. QUIZ 4
12) Consider the linear system
a) Find the general solution. ˚ −4 3 X
X =
2 −3 b) Use your solution to create a phase portrait for the system. _ _ _ _3 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _1 _ _ _ _ 3
 2
 1
 3
 4
 _ _ _ _1 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _3 _ _ _ _ _ _ _ _4 _ _ _ _ 1
 2
 13) Consider the linear system
a) Find the general solution. 4
3
˚
X =
X
−2 −3 b) Use your solution to create a phase portrait for the system. _ _ _ _3 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _1 _ _ _ _ 3
 2
 1
 3
 4
 _ _ _ _1 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _3 _ _ _ _ _ _ _ _4 _ _ _ _ 1
 2
 14) Consider the linear system
a) Find the general solution. −5 10
˚
X =
X
−10 −5 b) Use your solution to create a phase portrait for the system. _ _ _ _3 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _1 _ _ _ _ 3
 2
 1
 3
 4
 _ _ _ _1 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _3 _ _ _ _ _ _ _ _4 _ _ _ _ 1
 2
 15) Consider the linear system
a) Find the general solution. ˚ −5 −3 X
X =
3
1 b) Use your solution to create a phase portrait for the system. _ _ _ _3 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _1 _ _ _ _ 3
 2
 1
 3
 4
 _ _ _ _1 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _3 _ _ _ _ _ _ _ _4 _ _ _ _ 1
 2
 16) Create the phase portrait for the nonlinear system given
x =xy
˚
y = x1
˚
_ _ _ _3 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _1 _ _ _ _ 3
 2
 1
 3
 4
 _ _ _ _1 _ _ _ _ _ _ _ _2 _ _ _ _ _ _ _ _3 _ _ _ _ _ _ _ _4 _ _ _ _ a
17) If c b d 1
 2
 gives complex eigenvalues, what must be true about the product b×c ? Project
18) Describe in words what you did for your project. ...
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 Spring '08
 Hunter
 Partial differential equation, Gallon

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