MATH 202 Final Common Final, Spring 2014
American University of Beirut · Differential Equations · 15 pages · Prof. Wissam Raji
Showing the first 10 of 15 pages. Download the PDF to see all of it.
More MATH 202 files
Show the text of this final exam
American University of Beirut
Math 202-Differential Equations
Spring 2014-N. Nahlus, W. Raji, H. Yamani, M.
Kobeissi, K. Aziziheris
Final Exam- Time: 120 minutes
May 14, 2014
Your Name:......................... and ID:...................
Grades:
1/8
2/8
3/21
4/8
5/8
6/8
7/24
8/15
Total/100
Notes:
• Calculators are not allowed.
1
PLEASE CIRCLE YOUR SECTION:
Section 1(N. Nahlus, TTh 1230, Recitation F 1),
Section 2(N. Nahlus, TTh 1230, Recitation F 12),
Section 3(N. Nahlus, TTh 1230, Recitation F 8),
Section 4(N. Nahlus, TTh 1230, Recitation F 9),
Section 5(N. Nahlus, TTh 2 Recitation W 8),
Section 6(N. Nahlus, TTh 2 Recitation W 12),
Section 7(N. Nahlus, Th 2 Recitation W 11),
Section 8(N. Nahlus, Th 2 Recitation W 2).
Section 9(H. Yamani, MWF 10, Recitation Th 3:30),
Section 10(H. Yamani, MWF 10, Recitation Th 11),
Section 11(H. Yamani, MWF 10, Recitation Th 9:30),
Section 12(H. Yamani, MWF 10, Recitation Th 8),
Section 13(M. Kobeissi, MWF 2, Recitation F 12),
Section 14(M. Kobeissi, MWF 2, Recitation F 3),
Section 15(M. Kobeissi, MWF 2, Recitation F 4),
Section 16(M. Kobeissi, MWF 2, Recitation F 5),
Section 17(K. Aziziheris, MWF 8, Recitation M 2),
Section 18(K. Aziziheris, MWF 8, Recitation F 2),
Section 19(K. Aziziheris, MWF 8, Recitation F 4),
Section 20(W. Raji, MWF 1, Recitation M 8),
Section 21(W. Raji, MWF 1, Recitation M 9),
Section 22(W. Raji, MWF 1, Recitation M 2),
Section 23(W. Raji, MWF 1, Recitation M 5).
2
Usual Laplace formulas and rules:
f(t)
F(s)
1
1
s
t
1
s2
tn
n!
sn+1
eat
1
s−a
cos(kt)
s
s2+k2
sin(kt)
k
s2+k2
- L{f(t)} =
� +∞
0
e−stf(t)dt
- L{f(n)t)} = snF(s) − sn−1f(0) − sn−2f′(0) − . . . − f(n−1)(0)
- L{eatf(t)} = F(s − a)
- L{f(t − a)U(t − a)} = e−asF(s)
- L{U(t − a)} = e−as
s
- L{g(t)U(t − a)} = e−asL{g(t + a)}
- L{tnf(t)} = (−1)n dn
dsn F(s)
- f ∗ g =
� t
0 f(τ)g(t − τ)dτ
- L{f ∗ g} = L{f}L{g}
- L{δ(t − t0)} = e−st0
3
Problem 1.
(8 pts) Use Gauss divergence Theorem carefully to
find the outward flux
� �
S F · ndσ of the vector field
−→
F =
1
�
x2 + y2 + z2 (2x−→i + y−→j + 3z−→k
�
x2 + y2 + z2)
across the surface ρ = 3.
4
Problem 2. (8 pts) Find the area of the surface cut form the plane
x+2y +2z = 5 by the cylinder whose walls are x = y2 and x = 2−y2.
5
Problem 3. (21 pts, each part 7 pts) Solve the following DE’s:
•
xdy
dx + 2y = x2y2, y(1) = 1/5
6
•
dy
dx =
�
1 + (2x + y)3
(2x + y)2
− 2
7
•
x2y′′ − 2xy′ + 2y =
x3
√
1 − x2
8
Problem 4.
(8 pts) Use Frobenius method to find ONE series
solution (corresponding to the larger root) of the equation
x(x − 1)y′′ + 3y′ − 2y = 0
about the regular singular point x = 0. (Hint: Only find the first
three non-zero terms)
9
Problem 5. (8 pts) Use Laplace Transform and solve the following
DE:
y′′ + 2y′ + 10y = δ(t − 2) + 13et, y(0) = 0, y′(0) = 13
10
Problem 6.
(8 pts) Use Laplace Transform and solve the given
equation:
−3
8 (1 + t − f(t)) =
� t
0
f(τ)(t − τ)3dτ
11
Problem 7. (24 pts, each part 8 pts) Answer the following ques-
tions:
1. Find f(t) given that
F(s) = (
s
s2 + 25)(4),
where the exponent (4) is the fourth derivative.
12
2. Use the definition of Laplace Transform to find the value of
� ∞
0
e−3t cos(4(t − 5))U(t − 5)dt
13
3. Find the Laplace of
f(t) = 1 − cos t
t
14
Problem 8. (15 pts) Find the general solution of the given system:
X′ = AX,
where
A =
7
−4
0
1
0
2
0
2
7
15