Differential Equations Final Practice Exam
Differential Equations Final Practice Exam
1. (Final Spring 1996 Problem 2) Find the general solution of the following differential equations.
a) [pic] b) [pic]
Solutions
a) Integrating Factor Method: [pic]
[pic]. Let [pic]
[pic]
b) First find the homogeneous solution and then find the particular solution to find the general solution. Since the roots of the characteristic equation equals are the same as exponent in front of the exponential, you will need to add a factor of [pic] because of the repeated root of the homogeneous solution. This problem is basically the same as Example 6 on p. 234 with exception that you should find that A=1/2.
[pic]
2. (Final Spring 1996 Problem 3) Consider the differential equation [pic], [pic], [pic].
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?
Solutions
a) [pic]. Check if well defined at initial value. [pic]. So it is well defined. Check if [pic] is well defined at the initial values [pic], [pic]. Since both [pic]and [pic]are both well defined at the initial values, a unique solution exists surrounding the value x=0. The only restriction is on x because sec x is defined on the interval [pic].
b) Homogeneous equation
[pic]
[pic]
I will use the method as outlined on p.419 takes a little less work. Although you can use this method on the exam, (WARNING) I heard that no partial credit will be given if you make a mistake!!! So try the problem as professors taught section 8.2 to give you practice to do it either way. It really depends how good you are at memorizing.
Solving [pic]
[pic]
[pic]
[pic]. Let [pic][pic].
[pic]
[pic]
General solution [pic]
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?
Solutions
a) Salt inflow is 0
Salt outflow is [pic]
[pic] [pic]
[pic]
b) [pic]
Using the initial condition, [pic]
c) [pic]
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 [pic].
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.
Solutions
a) [pic]represents mass times acceleration
[pic] represents the damping force (friction)
u represents the restoring force
10sin t represents external “driving” forces in this case the crank
b) [pic]
Homogeneous solution:
[pic]
Particular solution:
Guess for a particular solution [pic]
Solving for coefficients A and B
[pic]
[pic]
Thus [pic]
General solution: [pic]
d) At this point, the solution to part c I don’t know… not the same as calculating SHO where we can calculate the amplitude using the formula on p.196. I would not worry about this problem since you were not asked about this before.. Just make sure you know how to find the amplitude of SHO.
At time t=0, [pic]
At time t=2 [pic]
Period of the oscillation is [pic]
5) (Final Spring 1996 Problem 10)
Solutions
a) [pic]
[pic]
Since alpha = 0 (i.e. the eigenvalue is purely imaginary), the solution is a center and is stable.
b)
[pic]
Since the eigenvalues are real distinct roots, with one eigenvalue positive and the other eigenvalue negative, the solution is unstable.
Eigenvectors [pic]
[pic]
c)
[pic]
Since the eigenvalues are complex numbers and alpha=-1 is less than 0. The solution is asymptotically stable.
In calculating the eigenvectors, we will utilize the fact that the eigenvectors are complex conjugates of each other since the eigenvalues are complex conjugates.
[pic]
b) (Final Fall 1996 Problem 4)
Solutions
a) [pic], [pic]
Hence the equilibria (1,1) and (-1,1)
b) [pic]
At equilibrium point (1,1), [pic]
Since alpha =1 is greater than 0, the solution is unstable , repelling spiral point.
At equilibrium point (-1,1), [pic]
Since the eigenvalues are real distinct roots and one eigenvalue is positive with the other eigenvalue negative, the equilibrium point is a saddle and the solutions are unstable.
c) (Final Fall 1998 Problem 6) For the linear system of differential equations [pic], [pic].
Solutions
a) [pic]
b) [pic]
[pic]
eigenvalues [pic]
c) [pic]
[pic]
[pic] are the eigenvectors (utilizing the fact the eigenvectors will be complex conjugates because the eigenvalues are complex numbers)
Note that any multiple (where r can be any complex number) would be an acceptable solution.
a. Since [pic], the general solution is equal to
[pic]
e) [pic]
[pic],[pic]
[pic]
[pic] and [pic]
f) [pic] means trajectories are closed loops. See p. 347 of the text of how to Figure 6.3.3 and Example 4 to draw the phase-plane.
d) (Final Fall 1998 Problem 7) For each of the following answer TRUE or FALSE.
Solutions
a) True
b) False
c) False
d) [pic][pic]
[pic]
Since both eigenvalues are positive, the solutions are moving away from the equilibrium point. False
e) [pic][pic]
[pic]
Since the exponents of the exponentials in the proposed solution do not agree with the eigenvalues found, the proposed set is not a fundamental set of solutions.
False
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