Practice Final Exam
Intro. DEs
Spring 2005
• Study Practice fianl exam and Midterms Equally
•Main focus
1. Linear first-order DE (page 46) and its application (mixture problem,page 51)
2. separable DE (page31) and its application (population model, sec
2.1,2.2)
3. n-th order linear DEs (homogeneous and non-homogeneous) and
mass-spring system.
4. Linear system and application (salt problems and mechanical
vibrations, see Midterm 2): two methods -Eigenvector method and
Laplace Transform method.
5. Almost linear system (type and stability of critical points)
6. DEs with discontinuous input data (Laplace Transform method)
1
2
1. A 100-gallon tank initially contains 80 lb of salt dissolved in 80
gal of water. Brine containing 2 lb (per gal) of salt flows into the tank
at the rate of 4 gal (per min), and the well-stirred mixture flows out of
the tank at the rate of 3 gal(per min). Find the amount of salt x(t) at
time t.
3
2. Let P (t) be a population function, and let β(t), α(t) denote the
birth rate, death rate, respectively.
(a) Derive the following equation:
dP
= (β(t) − α(t))P
dt
(b) Solve P (t) if β(t) = 0.01, α(t) = 0.001P (t) and P (0) = 9.
(c) Sketch the graph of the solution found in (b)
4
3. (a) Find the general solution of the equation
(D 3 + D − 10)(D 2 + 6D + 13)2 y = 0
(b) Solve the following mass and spring system
x′′ + 25x = 90 cos 4t,
x(t) = 0, x′ (0) = 90
5
4. (a) prove
L−1 {
(s2
1
s
}=
t sin kt
2
2
+k )
2k
(b) Let f (t) = cos 2t if 2π < t ≤ 4π and f (t) = 0, otherwise. Using
the result of (a) and Laplace Transform, solve the following equation
x′′ + 4x = f (t),
x(0) = x′ (0) = 0
6
5. Using Laplace Transform Method, find the solution of the system
x′′ + 3x − y = 0 y ′′ − 2x + 2y = 0
subject to the initial conditions
x(0) = y(0) = 1,
x′ (0) = y ′ (0) = 0.
7
6. Find the solution of

−1
′

1
X =
0
the system

0
1
−2 0  X,
2 −1
 
0

X(0) = 3
0