Test #1

 

Name and Signature of Student _____________________________


Signed by Proctor or Attendant, with Current Date and Time: ______________________


If picture ID has been matched with student and name as given above, Attendant please sign here: _________


Instructions:


Test is to be taken without reference to text or outside notes.

 

No calculator is necessary for this test.  However a Casio fx-260 calculator may be used.  No other calculator is permitted for this test.


Test is to be taken on blank paper or testing center paper.


There is no time limit, but the test is to be taken in one sitting. If taken at a testing center other than the VHCC center, a time limit not under 2 hours may be imposed by the center.  If so the time limit should be specified in a brief note on this page.


Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken.

 

Directions for Student:


Completely document your work. Show all steps and explain all reasoning.


Unless test is to be faxed or sent as PDF or other electronic means, please write on one side of paper only, and if possible staple test pages together.

 

Be sure to write in dark pencil or ink, especially if the test is to be scanned or faxed.

 

[I have inserted notes on most of the problems indicating a possible strategy for solution.

I'll be glad to critique your solutions and answer your questions, which you can submit using the Question Form.]

1. Solve the equation y ' - 2 y = -1 for y(1) = 2.

[use integrating factor e^-(2t) to get (y e^(-(2t) ) ' = e^(-2 t) with solution implicit in y e^(-2 t) = 1/2 e^(-2t) + c so that y = 1/2 + c e^(2 t); evaluate c and check that solution works]

2. The temperature of a room approaches the outdoor temperature at a rate proportional to the difference between the two temperatures. The outdoor temperature is -10 Celsius, and the initial room temperature is 20 Celsius. The average rate at which temperature changes during the first 30 minutes is -.2 Celsius / minute.

Write the differential equation for this situation, and use it along with the given conditions to find the temperature of the room as a function of clock time.

Do your best to answer the following question for this room:  Any reasonable attempt will be given at least partial credit.  A complete solution will get some extra credit.  An acceptable full-credit answer would include either of the following:

·         A correct differential equation which could be solved to answer the question.

·         A good graphical description of the direction field of the equation.

[the temperature T approaches room temperature at a rate proportional to the temperature relative to room temperature, so

dT/dt = k (T - (-10 C) ).

Solve this equation, leaving k as an unknown parameter.

Find the temperature after 30 minutes, and use this along with your solution function to evaluate k.

Plot the resulting curve along with your direction field and verify that the curve and the field are consistent.]

A soft drink is taken from the refrigerator at 3 Celsius, and set in the room.  If the room had stayed at 20 Celsius, the temperature of the soft drink would have reached 15 Celsius after 40 minutes.  What will be the temperature of the room after 40 minutes, and what will be the temperature of the soft drink at this time?

3. Find the exact equation corresponding to dF = 0, where F(t, y) = sqrt(t) * y^3.  Explain how you would have solved this equation had you not been given the function F(t, y), and apply the condition y(0) = 1 to find a specific solution.

[dF = F_x dx + F_y dy, so the equation dF = 0 would be F_x dx + F_y dy = 0.  The partial derivatives of the F function comprise the N and M functions of the exact equation.  The equation must pass the test for exactness, since F_x y = F_y x for any differentiable function F(x, y).]

4. A swimmer pushes off the side of a pool and glides, slowing to a speed of 10 centimeters /second in 5 seconds after her feet lose contact with the wall, during which time she travels 300 centimeters.  Assuming that the net force acting on the swimmer is proportional to her velocity, so that her acceleration is proportional to her velocity, what are her initial velocity and the drag constant.  This can be solved without using her mass, but if you wish you may assume a reasonable mass.

[The force on the swimmer is of the form F = - c v so the m dv/dt = - c v, or m v dv/dx = - c v.  These equations can be solved to get v(t), or v(x), as appropriate to the given information.]

5. Find the general solution of the equation y ' + t^2 / y - 4 y  = 0.

[This can be rearranged into the standard form of a Bernoulli equation and solved accordingly.]

6. Solve the equation y ' - 8 sin(2 t) = 6 y.

[This equation can be rearranged into the form of a linear equation and solved using an integrating factor.]

7.  A 1% saline solution flows into a 100-gallon tank initially full of 4% saline solution.  A thoroughly mixed solution flows out of the tank at the same rate as the inflow.  What is the rate of inflow if after 1 hour the concentration in the tank is 3% saline?

8.  Find the solution of the equation y ' = y^2 x / 3 for initial condition y(0) = 1.  Evaluate your solution at x = 1.

Use an Euler approximation with step 1/4 to estimate the solution at x = 1. 

Your estimate will either be high or low.  Which is it and why is it so?

 

 


 

 


 

 

Test #2

 

Name and Signature of Student _____________________________


Signed by Proctor or Attendant, with Current Date and Time: ______________________


If picture ID has been matched with student and name as given above, Attendant please sign here: _________


Instructions:

 

Test is to be taken without reference to text or outside notes.

 

No calculator is necessary for this test.  However a Casio fx-260 calculator may be used.  No other calculator is permitted for this test.


Test is to be taken on blank paper or testing center paper.


There is no time limit, but the test is to be taken in one sitting. If taken at a testing center other than the VHCC center, a time limit not under 2 hours may be imposed by the center.  If so the time limit should be specified in a brief note on this page.


Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken.

 

Directions for Student:


Completely document your work. Show all steps and explain all reasoning.


Unless test is to be faxed or sent as PDF or other electronic means, please write on one side of paper only, and if possible staple test pages together.

 

Be sure to write in dark pencil or ink, especially if the test is to be scanned or faxed.

 

1. The equation y '' + 5 y ' + 6 y = 0 has solutions y = e^(-3 t) and y = e^(-3 t + 2). Do these functions form a fundamental set on (-infinity, infinity)?

2. An unforced LRC circuit has equation Q '' + R Q ' + Q / ( L C) = 0. If C = .0072 Farads and L = .02 Henries, what is the form of the general solution? If the circuit is driven by voltage V(t) = 4 volts * sin(100 rad/s * t), what is the general solution for which Q(0) = 0 and Q ' (0) = 0? Describe the behavior of the circuit in its transient stage (near t = 0) and in in the long term.

3. Solve the equation y '' + y ' - 2 y = 0 with y(0) = 1 and y ' (0) = 1.

4. Solve the equation y '' + 3 y ' + 5 y = 0 with y(0) = -1 and y ' (0) = 1.

5. A spring-and-dashpot system has mass 2 kg and force constant 800 N / m. For what drag constant is this system critically damped? If the critically damped system is given an initial velocity of 4 m/s at a position 20 cm from equilibrium, what is its maximum displacement from equilibrium? Note that a position 20 cm from equilibrium could be on either side of equilibrium; answer for both caes, and compare your results. Explain the comparison in terms of the behavior of the physical system.

6. Solve the equation y '' - 8 y ' + 15 y = t^3 with y(0) = 0 and y ' (0) = 2.


 

 

Test #3

 

Name and Signature of Student _____________________________


Signed by Proctor or Attendant, with Current Date and Time: ______________________


If picture ID has been matched with student and name as given above, Attendant please sign here: _________


Instructions:

 

Test is to be taken without reference to text or outside notes.

 

No calculator is necessary for this test.  However a Casio fx-260 calculator may be used.  No other calculator is permitted for this test.


Test is to be taken on blank paper or testing center paper.


There is no time limit, but the test is to be taken in one sitting. If taken at a testing center other than the VHCC center, a time limit not under 2 hours may be imposed by the center.  If so the time limit should be specified in a brief note on this page.


Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken.

 

Directions for Student:


Completely document your work. Show all steps and explain all reasoning.


Unless test is to be faxed or sent as PDF or other electronic means, please write on one side of paper only, and if possible staple test pages together.

 

Be sure to write in dark pencil or ink, especially if the test is to be scanned or faxed.

 

Mth 279 Test Version A, on systems of differential equations and Laplace Transforms

Test is to be taken using test center paper. No outside notes, calculator or computer may be used on the test.

1. Solve the system

y_1 ' = -2 y_1 + y_2

y_2 = y_1 - 2 y_2

with initial condition y(0) = 1.

2. The system y ' = A y, with A = [1,5;2,-2], has solutions y_1 = e^(-4 t) * [ 1, -1] , y_2 = e^(3 t) * [ 5,2 ].

Show that this solution set satisfies Abel's Theorem with t_0 = 0.

3. Find the propagator matrix for the system y ' = A y, where A is the matrix [-2, -2; 1, -4], and use it to find y(pi) if we know that y(0) = [0, 1].

4. Find the general solution of the system y ' = A y for A = [-6, 2; 2, -2 ]. Write the fundamental matrix psi(t) for this solution and show that psi ' (t) = A psi(t).

5. Use the definition of the Laplace transform to validate the following from the notes on Laplace Transforms at the end of this document:

"The function t f(t) has Laplace transform - F ' (s), where F(s) is the Laplace transform of f(t)"

6. Use the Laplace transform to solve the equation y ' + 4 y = cos(pi t) with y(0) = 1.

Notes on Laplace transforms

The function t f(t) has Laplace transform - F ' (s), where F(s) is the Laplace transform of f(t).

The function t^n f(t) has Laplace transform (-1)^n F[n](s) , where F[n](s) is the nth derivative of the Laplace transform of f(t).

The function f ' (t) has Laplace transform s F(s) - f(0), where F(s) is the Laplace transform of f(t)

The function f '' ( t) has Laplace transform s^2 F(s) - s f(0) - f ' (0), where F(s) is the Laplace transform of f(t).

The function integral(f(tau) dTau, tau from 0 to t) has Laplace transform 1/s F(s), where F(s) is the Laplace transform of f(t).

The function e^(a t) f(t) has Laplace transform F(s - a), where F(s) is the Laplace transform of f(t)

The function f(t-a) * h(t-a) has Laplace transform e^(-as) F(s), where F(s) is the Laplace transform of f(t) and h(t) is the Heaviside function (equal to 0 for t < 0, equal to 1 for t >= 0)

The function sin(omega t) has Laplace transform omega / (s^2 + omega^2).

The function cos(omega t) has Laplace transform s / (s^2 + omega^2).

The function h(t) has Laplace transform 1/s, where h(t) is the Heaviside function (see previous note on Heaviside function).