Math 250 NAME:Fall 2010Final Exam ID No:

SECTION:

This exam contains 17 problems on 13 pages (including this title page) for a total of 150points. There are 10 multiple-choice problems and 7 partial credit problems. In order toobtain full credit for the partial credit problems, all work must be shown.

NO CALCULATORS, NOTES OR BOOKS ARE ALLOWED. All cellphonesand music players, or any other electronic device must be put away.

Problem Score Possible Points

1 6

2 6

3 6

4 6

5 6

6 6

7 6

8 6

9 6

10 8

11 7

12 9

13 13

14 14

15 18

16 20

17 7

Total 150

MATH 250 – Final Exam – Fall 2010

Multiple Choice Section

1. (6 points) Which of the following is the general solution of the equation:

y′′ + 3y′ + 2y = 0?

(a) y = c1 sin(2t) + c2 cos(t);

(b) y = c1e2t + c2e

t;

(c) y = c1e−2t + c2e

−t;

(d) None of the above.

2. (6 points) Which of the following second order differential equations is linear andhomogeneous?

(a) t2y′′ + ty′ + (sin t)y = 0;

(b) (1 + y2)y′′ + ty′ + (sin t)y = 0;

(c) t2y′′ + ty′ + (sin t)y = ln t;

(d) None of the above.

Page 2 of 13

MATH 250 – Final Exam – Fall 2010

3. (6 points) Which of the following is the general solution of the equation: y′ − y = e3t?

(a) y = t3 + Cet;

(b) y = te3t + Cet;

(c) y = 1

2e3t + Ce−t;

(d) y = 1

2e3t + Cet.

4. (6 points) Consider g(t) = t2 − 3u1(t) + u3(t)− tu5(t). What is g(3)− g(0)?

(a) 6

(b) 7

(c) 4 + e6

(d) 3

Page 3 of 13

MATH 250 – Final Exam – Fall 2010

5. (6 points) Find the solution of the initial value problem:

y′ =3− ex

4 + 2y, y(0) = 1

(a) y = −2 +√3x− ex + 10;

(b) y = −2 +√3x2 − ex + 10;

(c) y = 4−√3x− ex + 10;

(d) y = −4 −√3x− ex + 10.

6. (6 points) Which of the following is the Laplace transform of f(t) = tu3(t)?

(a) F (s) = e−3s;

(b) F (s) = e−3s

s2;

(c) F (s) = e−3s( 1

s2+ 3

s);

(d) None of the above.

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MATH 250 – Final Exam – Fall 2010

7. (6 points) Using the method of undetermined coefficients, a particular solution of

y′′ − 4y′ + 4y = te2t + t2

is of the form

(a) (At+B)e2t + Ct2 +Dt+ E;

(b) (At2 +Bt)e2t + Ct2 +Dt+ E;

(c) (At3 +Bt2)e2t + Ct2 +Dt + E;

(d) Ate2t + Ct2 +Dt+ E.

8. (6 points) An interval, where the solution to the following initial value problem existsand is unique, is:

(t sin t)y′ + t2 cos t =y

t2 − 4, y(−3) = 1.

(a) (−∞,−2);

(b) (−2, 2);

(c) (2, ∞);

(d) (−π,−2).

Page 5 of 13

MATH 250 – Final Exam – Fall 2010

9. (6 points) Consider the system

x′ =

(

0 11 0

)

x.

Which of the following is the direction field of the above system?

(A) (B)

(C) (D)

10. (8 points) Consider the system

x′ =

(

0 −1k 2

)

x.

Which of the following statement is false:

(a) The origin is a spiral point if k > 1

(b) The origine is a center when k = 1

(c) The origin is a saddle point when k = −3

(d) All saddle points of this system are unstable.

Page 6 of 13

MATH 250 – Final Exam – Fall 2010

Partial Credit Section

11. (7 points) Find an integrating factor for the following differential equation:

ty′ − (t + 2)y = 2 tan(3t)

12 (9 points) Consider the autonomous differential equation y′ = y2(y2 − 25). List theequilibrium points and determine their stability. No explanations required.

equilibrium points stability

Page 7 of 13

MATH 250 – Final Exam – Fall 2010

13. (13 points) Consider the differential equation

t2y′′ − 3ty′ + 4y = 0, t > 0.

(a) (5 points) Use Abel’s theorem to compute the Wronskian of a fundamental set ofsolutions of this differential equation without solving it.

(b) (8 points) Use the Wronskian computed above to compute the general solution ofthe above differential equation, given that one solution is y1(t) = t2.

Page 8 of 13

MATH 250 – Final Exam – Fall 2010

14. (14 points) Consider the initial value problem:

y′′ + y = δ(t− π) , y(0) = 0, y′(0) = 0.

Use the Laplace transform to solve the given initial value problem by proceeding asfollows.

(a) (7 points) Find the Laplace transform Y (s) = L{y} of the solution of the giveninitial value problem.

(b) (7 points) Compute the inverse Laplace transform to solve the initial value problem.

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MATH 250 – Final Exam – Fall 2010

15. (18 points)

(a) (6 points) Transform the given system of differential equations below into a singleequation of second order. (Do not solve it!)

x′

1 = 2x1 − 3x2

x′

2 = x1 − x2

(b) (6 points) Transform the given initial value problem for the single differentialequation of second order into an initial value problem for two first order equations.(Do not solve it!)

y′′ + 3y′ + y = et, y(0) = −1, y′(0) = 2.

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MATH 250 – Final Exam – Fall 2010

(c) (6 points) (Problem 15, continued)

Consider the linear system x′ = Ax, where A is a 2×2 matrix with eigenvectors

(

11

)

and

(

12

)

corresponding to the eigenvalues r1 = −1 and r2 = 2, respectively. Find a

fundamental matrix Φ of the system such that Φ(0) = I.

Page 11 of 13

MATH 250 – Final Exam – Fall 2010

16. (20 points) Consider the initial value problem

x′ =

(

2 1−1 4

)

x ; x(0) =

(

11

)

.

(a) (4 points) Find the eigenvalue(s) and the eigenvector(s) of the above 2×2 matrix.

(b) (6 points) Find a fundamental set of two solutions for the system.

(c) (5 points) Find the solution of the above initial value problem.

(d) (5 points) Classify the origin and determine the stability of the system.

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MATH 250 – Final Exam – Fall 2010

17. (7 points)

(a) (3 points) Write down the equation that describes the Euler method for numer-ically approximating the solution to an initial value problem

dy

dt= f(t, y) , y(t0) = y0

with step size h.

Your answer: yn+1 =

(b) (4 points) Consider the initial value problem

dy

dt= sin t+ y2 + 1, y(0) = 1.

Use Euler’s method with step size h = 0.1 to find the approximate value of thesolution of the initial value problem at t = 0.1.

Page 13 of 13