MATH 251 Final Examination December 16, 2015 Student

MATH 251

Final Examination

December 16, 2015

FORM A

Name:

Student Number:

Section:

This exam has 17 questions for a total of 150 points. In order to obtain full credit forpartial credit problems, all work must be shown. For other problems, points might bededucted, at the sole discretion of the instructor, for an answer not supported by areasonable amount of work. The point value for each question is in parentheses to the right ofthe question number. A table of Laplace transforms is attached as the last page of the exam.

You may not use a calculator on this exam. Please turn off and put away yourcell phone and all other mobile devices.

1

through

12: (72)

13: (10)

14: (16)

15: (18)

16: (16)

17: (18)

Total:

Do not write in this box.

MATH 251 FINAL EXAMINATION Form A December 16, 2015

1. (6 points) Which of the following equations is a nonlinear third order ordinary differentialequation?

(a) y′′′ + 2y′′ + 3t2y′y = 0

(b) y3 + 7y2 − 9y = 12

(c) y′′′ + 2ty′′ − ety = sin(t)

(d) (y′′)3/2 + t2 + y2 = 0

2. (6 points) Consider the initial/boundary value problems below. Which one is certain to havea unique solution for every value of α?

I y′′ + αy = 0, y(0) = 0, y(π) = 0.

II t2y′′ + α2y′ = tan t, y(π) = 0, y′(π) = −α.

(a) I only.

(b) II only.

(c) Both I and II.

(d) Neither.

Page 2 of 13


3. (6 points) Which of the following functions is a solution of the nonhomogeneous linear equation

y′′ − 4y′ − 5y = 12e5t + 130 sin(5t) ?

(a) y = 3e−t + 5 cos(5t)

(b) y = e−t + e5t

(c) y = 6 cos(5t)

(d) y = 2(1 + t)e5t + 2 cos(5t)− 3 sin(5t)

4. (6 points) Consider the autonomous equation

y′ = (y2 − 1)(y + 2).

Given the initial conditions y(2015) = α, find all possible values of α such that limt→∞

y(t) is

finite.

(a) −∞ < α ≤ 1

(b) −2 ≤ α ≤ 1

(c) −∞ < α <∞

(d) −2 < α < 1

Page 3 of 13


5. (6 points) Which equation below has y = 5−√

3et+2 as one of its solutions?

(a) y′′ − 6y′ + 5y = 1

(b) y′′ + y′ = 0

(c) y′′′ − y′′ = 0

(d) y(4) − y = 0

6. (6 points) Find the Laplace transform L{u2(t)(t− 2)e−t}.

(a) F (s) =e−2s

(s+ 1)2

(b) F (s) = e−2s−2s− 1

(s+ 1)2

(c) F (s) =e−2s−2

(s+ 1)2

(d) F (s) =e−2s+2

(s+ 1)2

Page 4 of 13


7. (6 points) Find the inverse Laplace transform L−1{e−3s 2s2 + 3

s3 − s}.

(a) f(t) = u3(t)

(5

2e−t+3 +

5

2et−3 − 3

)

(b) f(t) = u3(t)

(5

2e−t+3 +

5

2et−3

)

(c) f(t) = u3(t)

(5

2e−t−3 − 5

2et+3 − 3

)

(d) f(t) = δ(t− 3)

(5

2et − 5

2et + 3

)

8. (6 points) Find the general solution of the linear system

x′ =

[1 20 2

]x.

(a) x(t) = C1et

[10

]+ C2e

2t

[01

]

(b) x(t) = C1et

[10

]+ C2e

2t

[21

]

(c) x(t) = C1e−t[

11

]+ C2e

−2t[

23

]

(d) x(t) = C1e−t[

01

]+ C2e

−2t[

21

]

Page 5 of 13


9. (6 points) Given that the point (1, 0) is a critical point of the nonlinear system of equations

x′ = y2 + 2xy + 3yy′ = xy2 + 2xy + x− 1

.

The critical point (1, 0) is an

(a) unstable spiral point.

(b) unstable saddle point.

(c) asymptotically stable spiral point.

(d) asymptotically stable node.

10. (6 points) Consider the two linear partial differential equations.

I uxx − 3uxt + ut = 0

II uxx + 5utt = u

Use the substitution u(x, t) = X(x)T (t), where u(x, t) is not the trivial solution, and attemptto separate each equation into two ordinary differential equations. Which statement below istrue?

(a) Both equations can be separated.

(b) Only I can be separated.

(c) Only II can be separated.

(d) Neither equation can be separated.

Page 6 of 13


11. (6 points) Find the steady-state solution, v(x), of the heat conduction problem with nonho-mogeneous boundary conditions:

9uxx = ut, 0 < x < 4, t > 0u(0, t) + ux(0, t) = 3, u(4, t)− 2ux(4, t) = 8,u(x, 0) = 3x+ 2.

(a) v(x) =5

4x+ 3

(b) v(x) = 5x− 2

(c) v(x) = 3x+ 2

(d) v(x) = 8x+ 3

12. (6 points) Each graph below shows a single period of a certain periodic function. Whichfunction will have a Fourier series that contains at least one nonzero cosine term and at leastone nonzero sine term?

(a)

−π −π2

π2

π

−1

1

(b)

−π −π2

π2

π

−1

1

(c)

−π −π2

π2

π

−1

1

(d)

−π −π2

π2

π

−1

1

Page 7 of 13


13. (10 points) True or false:

(a) The equationdy

dx=

ex−y

x2 − y2is a separable equation.

(b) Suppose f(t) = L−1{ s

(s− 1)3}. Then f(2) = 4e2.

(c) Every nonzero solution of the linear system x′ =

[2 3−2 1

]x moves away, unbounded,

from (0, 0), as t→∞.

(d) Every Fourier series of an even periodic function has 0 as its constant term.

(e) Using the formula u(x, t) = X(x)T (t), where u(x, t) is not the trivial solution, the bound-ary conditions ux(0, t) = 0 and u(5, t) = 0 can be rewritten as X ′(0) = 0 and X(5) = 0.

Page 8 of 13


14. (16 points) Consider the two-point boundary value problem

X ′′ + λX = 0, X(0) = 0, X ′(7) = 0.

(a) (12 points) Find all positive eigenvalues, and their corresponding eigenfunctions, of theboundary value problem. You must show all supporting work to you answer.

(b) (4 points) Is λ = 0 an eigenvalue of this problem? If yes, find its corresponding eigenfunc-tion. If no, briefly explain why it is not an eigenvalue.

Page 9 of 13


15. (18 points) Let f(x) =

{−1, 0 6 x < 2,

3− x, 2 6 x < 3,

(a) (4 points) Consider the odd periodic extension, of period T = 6, of f(x). Sketch 3 periods,on the interval −9 < x < 9, of this function.

(b) (4 points) To what value does the Fourier series of this odd periodic extension convergeat x = −2? At x = 9?

(c) (4 points) Consider the even periodic extension, of period T = 6, of f(x). Sketch 3periods, on the interval −9 < x < 9, of this function.

(d) (3 points) Finda02

, the constant term of the Fourier series of the even periodic function

described in (c).

(e) (3 points) Which of the integrals below can be used to find the Fourier cosine coefficientsof the even periodic extension in (c)?

1. an = 13

(−∫ 20 cos nπx3 dx+

∫ 32 (3− x) cos nπx3 dx

)2. an = 2

3


∫ 32 (3− x) cos nπx3 dx

)3. an = 1

6


∫ 32 (3− x) cos nπx3 dx

)4. an = 1

3

(∫ −2−3 (3− x) cos nπx3 −

∫ 2−2 cos nπx3 dx+

∫ 32 (3− x) cos nπx3 dx

)

Page 10 of 13


16. (16 points) Suppose the temperature distribution function u(x, t) of a rod is given by theinitial-boundary value problem

4uxx = ut, 0 < x < 10, t > 0,u(0, t) = 0, u(10, t) = 0, t > 0,u(x, 0) = −3 sin

(πx5

)+ 6 sin

(πx2

), 0 < x < 10.

(a) (12 points) State the general form of its solution. Then find the particular solution of theinitial-boundary value problem.

(b) (2 points) What is limt→∞

u(x, t)?

(c) (2 points) Suppose the boundary conditions were changed to u(0, t) = 20 and

u(10, t) = 0, respectively. What is limt→∞

u(3, t) in this case?

Page 11 of 13


17. (18 points) Suppose the displacement u(x, t) of a piece of flexible string is given by the initial-boundary value problem

4uxx = utt, 0 < x < 3, t > 0u(0, t) = 0, u(3, t) = 0,u(x, 0) = 0,ut(x, 0) = 5 cos(πx) + 2.

(a) (2 points) TRUE or FALSE: At t = 0, the string is at rest (i.e., having zero initial velocity).

(b) (3 points) When t = 0, what is the displacement of the string at the midpoint, x = 32?

(c) (5 points) In what specific form will the general solution appear?

(1) u(x, t) =∞∑n=1

Cn cos

(2nπt

3

)sin(nπx

3

), (2) u(x, t) =

∞∑n=1

Cn sin

(2nπt

3

)cos(nπx

3

),

(3) u(x, t) =

∞∑n=1

Cn cos

(2nπt

3

)cos(nπx

3

), (4) u(x, t) =

∞∑n=1

Cn sin

(2nπt

3

)sin(nπx

3

).

(d) (4 points) TRUE or FALSE: The coefficients of the solution in part (c) above can be foundusing the integral

Cn =1

nπ

∫ 3

0(5 cos(πx) + 2) sin

(nπx3

)dx

(e) (2 points) TRUE or FALSE: The boundary conditions indicate that the string is securelyfixed and held motionless at both ends.

(f) (2 points) TRUE or FALSE: The quantity ux(2, 5) represents the velocity of the stringwhen t = 5 at the point x = 2.

Page 12 of 13

f(t) = L−1{F (s)} F (s) = L{f(t)}

1. 11

s

2. eat1

s− a

3. tn, n = positive integern!

sn+1

4. tp, p > −1Γ(p+ 1)

sp+1

5. sin ata

s2 + a2

6. cos ats

s2 + a2

7. sinh ata

s2 − a2

8. cosh ats

s2 − a2

9. eat sin btb

(s− a)2 + b2

10. eat cos bts− a

(s− a)2 + b2

11. tneat, n = positive integern!

(s− a)n+1

12. uc(t)e−cs

s

13. uc(t)f(t− c) e−csF (s)

14. ectf(t) F (s− c)

15. f(ct)1

cF(sc

)16. (f ∗ g)(t) =

∫ t

0f(t− τ)g(τ) dτ F (s)G(s)

17. δ(t− c) e−cs

18. f (n)(t) snF (s)− sn−1f(0)− · · · − f (n−1)(0)

19. (−t)nf(t) F (n)(s)

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MATH 251 Final Examination December 16, 2015 Student