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Differential Equations Final Practice Exam
1. (Final Spring 1996 Problem 2) Find the general solution of the following
differential equations.
2
dy 1
 y  ex
a)
b) y  2 y  y  e x
dx x
2. (Final Spring 1996 Problem 3) Consider the differential equation
1
y  y  sec x , y (0)  0 , y(0)  1 .
2
a) According to the theorem on existence and uniqueness, on what interval of
x is the solution guaranteed to exist and be unique?
b) Find the solution of the equation.
c) On what interval of x does the solution exist?
3. (Final Spring 1996 Problem 4) A tank contains 10,000 liters of a solution
consisting of 100 kg of salt dissolved in water. Pure rocky mountain spring water
is pumped into the tank at the rate of 10 liters/second. (The mixture is kept
uniform by stirring.) The mixture is pumped out of the tank at the same rate it
enters.
a) Set up the initial value problem for the amount of salt, S(t), in the tank at
time t.
b) Solve this problem for S(t).
c) How long will it be before only 50 kg of salt remain in the tank?
4. (Final Spring 1996 Problem 8) A damped spring is connected to a crank, which
forces the spring at a frequency of 1/sec. According to the laws of Newton and
Hooke, the governing equation is 2u  3u  u  10sin t .
a) Explain the physical meaning of each term in the equation.
b) Find the general solution of the equation.
c) If the spring is steadily cranked for 2 hours, determine the amplitude of the
solution.
5. (Final Spring 1996 Problem 10) For each of the following matrices A, consider
dx
 Ax . In each case, find the eigenvectors, eigenvalues, sketch the
the system
dt
phase portrait, and give the stability types (stable, unstable, or asymptotically
stable).
 0 4
1 1 
1 5 
a) A  
b) A  
c) A  



 0 3 
1 3 
 1 0 
6. (Final Fall 1996 Problem 4)
a) Determine all the critical points (equilibria) of the equations
dx
dy
 y  1,
 y 2  x2
dt
dt
b) For each of the critical points (equilibria) in a), determine whether they
are stable, asymptotically stable, or unstable.
7. (Final Fall 1998 Problem 6) For the linear system of differential equations
x1   x1  3x2 , x2  3x1  x2 .
a) Write the system in matrix-vector form x  Ax .
b) Find the characteristic polynomial and eigenvalues of the matrix A.
c) Find the corresponding eigenvectors of A.
d) Find two linearly independent real solutions, which form a basic set of
solutions for x  Ax .
e) Solve the initial value problems with x1 (0)  1 , x2 (0)  3 .
f) Draw the phase portrait or picture of solutions in the plane.
8. (Final Fall 1998 Problem 7) For each of the following answer TRUE or FALSE.
a) If det( A)  0 , then Ax  0 has infinitely many solutions.
b) If x1 , x2 ,..., xn are solutions of Ax  b , then c1 x1  c2 x2  ...  cn xn is a
solution of Ax  0 .
c) The linear system of equations Ax  0 where A is m n , n<m has only
trivial solutions.
 7 6
 x , all solutions approach the equilibrium point.
d) For x   
 2 6
 1 2  2
1
 1
 0


 
 
 
2t
t
e) For x    2 1  2  x , x1 (t )  e 1, x 2 (t )  e  0 , and x3 (t )   1 
 2 2  3
1
 1
1


 
 
 
is a basic (fundamental) set of solutions.