1. Given the plot of the function f(t) in one period, determine completely the Euler coefficients and write its Fourier series expansion. 3 f(t) -3 3 -3 12 0, 1 t t 2. Given the periodic function over the period [-1/2, n/2] f(x) = x |x| a. Plot f(x) on the interval [-, π] b. Is the function even or odd? c. Determine ao, an, bn. d. Setup the Fourier Series Expansion of f(x) 2 3 3. Given the periodic function below defined over a period, determine completely the Euler coefficients and write its Fourier series expansion. f(x) = -C < x <0 0
1. Given the plot of the function f(t) in one period, determine completely the Euler coefficients and write its Fourier series expansion. 3 f(t) -3 3 -3 12 0, 1 t t 2. Given the periodic function over the period [-1/2, n/2] f(x) = x |x| a. Plot f(x) on the interval [-, π] b. Is the function even or odd? c. Determine ao, an, bn. d. Setup the Fourier Series Expansion of f(x) 2 3 3. Given the periodic function below defined over a period, determine completely the Euler coefficients and write its Fourier series expansion. f(x) = -C < x <0 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I have qno4 ans but missing can u do again full accurate?? Qno4 do it
![1. Given the plot of the function f(t) in one period, determine completely the Euler
coefficients and write its Fourier series expansion.
3
f(t)
-3
3
-3
12
0,
1
t
t
2. Given the periodic function over the period [-n/2, π/2]
f(x) = x x
a. Plot f(x) on the interval [-, π]
b. Is the function even or odd?
c. Determine ao, an, bn.
d. Setup the Fourier Series Expansion of f(x)
2
3
3. Given the periodic function below defined over a period, determine completely the
Euler coefficients and write its Fourier series expansion.
f(x)
-C < x <0
0<x<c
4. Find the Fourier cosine expansion of f(x) = x²(3L-2x) on [0,L].
5. Solve the one-dimensional wave equation for a string of length 10 units, fixed on both
ends, with a = 3, if the string is initially at rest and with an initial displacement of
y(x,0)=x for 0<x< 10.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ab0fb29-2c34-45df-b1d3-efd8add8675a%2F1b54752f-7c73-4093-8ba6-abc95dd3d466%2F9atta6ya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Given the plot of the function f(t) in one period, determine completely the Euler
coefficients and write its Fourier series expansion.
3
f(t)
-3
3
-3
12
0,
1
t
t
2. Given the periodic function over the period [-n/2, π/2]
f(x) = x x
a. Plot f(x) on the interval [-, π]
b. Is the function even or odd?
c. Determine ao, an, bn.
d. Setup the Fourier Series Expansion of f(x)
2
3
3. Given the periodic function below defined over a period, determine completely the
Euler coefficients and write its Fourier series expansion.
f(x)
-C < x <0
0<x<c
4. Find the Fourier cosine expansion of f(x) = x²(3L-2x) on [0,L].
5. Solve the one-dimensional wave equation for a string of length 10 units, fixed on both
ends, with a = 3, if the string is initially at rest and with an initial displacement of
y(x,0)=x for 0<x< 10.
![(4) Given function f(x)=²(3L-2x) defined in [0,1].
The Fourier coefficients:
ao
an
||
2
²1 [ ² (3 La ² – 2x¹³) da
-
L
2
2132
2 LA
-
3
Lx³ 12¹1
L 2
L³
L
- ²/2 √² 2² (3L - 2x) cos (1²) da
dx
L
0
L
– ² (3Lz² - 2a³¹) --sin (¹72) + (6Lx − 6x²
=
-
nn
2413
n¹π4
241³
-(cos(nπ) - 1)
L²
n²72
((-1)" - 1)
Thus the Fourier Cosine series is
²+1 24²((-1)" - 1) cos (¹²)
-COS
ппх
(6L - 12x)-
L³
373
n³7](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ab0fb29-2c34-45df-b1d3-efd8add8675a%2F1b54752f-7c73-4093-8ba6-abc95dd3d466%2Fkuieb4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(4) Given function f(x)=²(3L-2x) defined in [0,1].
The Fourier coefficients:
ao
an
||
2
²1 [ ² (3 La ² – 2x¹³) da
-
L
2
2132
2 LA
-
3
Lx³ 12¹1
L 2
L³
L
- ²/2 √² 2² (3L - 2x) cos (1²) da
dx
L
0
L
– ² (3Lz² - 2a³¹) --sin (¹72) + (6Lx − 6x²
=
-
nn
2413
n¹π4
241³
-(cos(nπ) - 1)
L²
n²72
((-1)" - 1)
Thus the Fourier Cosine series is
²+1 24²((-1)" - 1) cos (¹²)
-COS
ппх
(6L - 12x)-
L³
373
n³7
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