SM 212 Final Examination
11 December 1996

1. (a)
Find the general solution to

(i) ,

(ii) ,

(b)
Solve the initial value problem .

2. (a)
A tank contains gallons of water with lbs. of salt dissolved in it. Saltwater with a concentration of lb/gal is pumped into the tank at a rate of gal/min. The well-stirred mixture is pumped out at the rate of gal/min. At what time does the tank contain the largest amount of salt? How much salt is in the tank at that time?

(b)
Use the improved Euler method with a step size of to approximate for the initial value problem.

3. Find the general solution to:

(a)
,

(b)
,

(c)
.

4. Use undetermined coefficients (or annihilators) to solve the initial value problem

5. (a)
An lb weight stretches a spring ft upon coming to rest at equilibrium. From equilibrium the weight is raised ft and released from rest. The motion of the weight is resisted by a damping force that in numerically equal to times the weight's instantaneous velocity.

(i) Find the position of the weight as a function of time.

(ii) What type of damping does this mass-spring system possess?

(b)
The position of a weight in a mass-spring system subject to an external force is given by .

(i) What are the amplitude and period of the steady-state part of the solution?

(ii) Write the transient part of the solution in the form .

(iii) Find the time past which the magnitude of the transient part of the solution is less than one-percent of that of the steady-state part of the solution.

6. (a)
Find:

(i) ,

(ii) ,

(iii) .

(b)
Find for the periodic function given over one period by

7. (a)
Find

(i) ,

(ii) .

(b)
Use the convolution theorem to find .

8. (a)
For the circuit in the circuit above show that the charge on the capacitor and the current in the right branch satisfy the system of differential equations

(b)
When the switch in the circuit is closed at time , the current is 0 amps and the charge on the capacitor is coulombs. With , , use Laplace transforms to find the charge on the capacitor.

9. Use eigenvalues/eigenvectors to solve the initial value problem

10. (a)
Find the Fourier sine series of the function

(b)
The temperature of a thin bar of length satisfies the following conditions

where is given in (10.a).

(i) Use your answer in (10.a) to find . Show all steps of the separation of variables process clearly. Write your answer in summation notation.

(ii) Use the first two non-zero terms of your answer in ((10)(b)1) to approximate .

Answers for SM 212 Final Examination
11 December 1996

1. a. i. , , , .

1. a. ii. , , , ,

1.b. , , , , , .

2. , , , gives . The IC , implies

2.b. , , , ,

3.a. , ,

3.b. , < , ,

3.c. , , (double roots), ,

4. , , , , , , , , , , , , , , ,

5.a. 1. , , slugs, , , , , , ,

5.a.2. critically damped.

5.b.1. Amplitude: , Period = ,

5.b.2. , ,

5.b.3. , , .

6.a.1. ,

6.a.2 ,

6.a.3

6.b.1 ,

7.a.1 , , , ,

7.a.2 , ,

7.b. Use the convolution theorem to find , .

8.a. ,

8.b. Substituting one DE into the other gives , so . Taking LTs: , , . Now take inverse LTs to get:
.

9. ,

10.a. , ,

b. , , .