Download Set #7

EGR 599 (Fall 2003)
_____________________
LAST NAME, FIRST
Problem set #7
1. Given
I
V
V
I
=L
 RI and
=C
 GV
x
t
x
t
In these equations, V(x, t) = voltage, I(x, t) = current, L = inductance, R = resistance, C =
capacitance, G = leakage to the ground. Show that V and I each satisfy
u
 2u
 2u
=
LC
+ (RC + LC)
+ RGu (Note: u denotes V or I)
2
2
t
x
t
2. Find the directional derivative of f(x, y) = 2x2 + y2 at x = 2 and y = 2 in the direction of h = 3i
+ 2j.
Ans: 8.875
3. Solve x
u
u
+y
= u; given that u = 3x on x + y = 1
x
y
Ans: u = 3x
4. If u = 3y2 when x = 0 on x = 0 solve
u
u
 (x  y  1)
= 3x(y  x)
x
y
Ans: u = 3(y  x)(x  1) + 3(y  x)e-x[(y  x)e-x + 1]
5. Use the change of variables  = x + ct,  = x  ct to transform the wave equation
2
 2u
 2u
2 u
=
c
into
= 0. Integrate the equation to obtain the solution u(x,t) = F(x + ct) +
t 2
x 2

G(x  ct).
6. Find the solution to the wave equation that satisfies the following boundary and initial
conditions
u(0,t) = 0, for t  0 and u(L,t) = 0, for t  0
u(x,0) = sin
Ans: u(x,t) = sin
2x
u
, for 0  x  L and
(x,0) = 0, for 0  x  L
t
L
2x
2ct
cos
L
L
7. Find a period of the function
(b) cos x
(a) cos x
2
(c) cos x
3
(d) cos x + cos 2x
Ans: (a) 2, (b) , (c) 3, (d) 2
8. Establish the orthogonality of the trigonometric functions over the interval [,].
9. Plot the Nth partial sums of the Fourier series for N = 1 and N = 5 for the following functions
over the interval    x  .
a)
f(x) = |x| =

4

2


1
 (2k  1)
k 0
2
cos( 2k  1) x 
1 term
3
2.5
|x|
2
1.5
1
0.5
0
-8
-6
-4
-2
0
x
2
4
6
8
2
4
6
8
5 terms
3.5
3
|x|
2.5
2
1.5
1
0.5
0
-8
-6
-4
-2
0
x
b)
f(x) = |sin x| =
2


4


1
 ( 2k )
k 1
2
1
cos( 2kx)
1 term
1.4
1.2
|sin(x)|
1
0.8
0.6
0.4
0.2
0
-8
-6
-4
-2
0
x
2
4
6
8
2
4
6
8
5 terms
1.4
1.2
|sin(x)|
1
0.8
0.6
0.4
0.2
0
-8
-6
-4
-2
 
10. Find the Fourier series of f(x) = a 1  
 

2
( 1) n 1
nx
Ans: f(x) = a + 4a 
cos(
)
2
3
p
n 1 ( n )
0
x
2
x 
  if  p  x  p.
p  

11. Find the Fourier series of f(x)
f(x) = 
Ans: f(x) =
2c
2c
(x  p/2) if 0  x  p and f(x) =
(x + p/2) if p  x  0 .
p
p
8c

2

1

 2k  1 cos[(2k + 1) p x]
k 0
12. Find the sine series expansion of x(x  1) on the interval 0 < x < 1.
Ans: x(x  1) =
8

3

1
 2k  1
k 0
3
sin[(2k + 1)x]
13. Find the sine series expansion of sin x on the interval 0 < x < 1.
Ans: f(x) = sin x
14. Find the sine series expansion of (sin x)(cos x) on the interval 0 < x < 1.
Ans: f(x) =
1
sin 2x
2