Answered: c) f(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

See similar textbooks

Related questions

Question

How would I do fourier (even) cosine series and fourier (odd) sine series on attached. I have to graph these too by hand and not sure how. This is piecewise function, Thanks so much

Expert Solution

Step 1

Solving Fourier cosine and sine series

$f (x) = \{\begin{cases} 0, 0 < x < 1 \\ 1, 1 < x < 2 \end{cases}$

Step 2

As $f (- x) = - f (x) s o f (x)$ is an odd function of x.

Hence, $a_{n} = 0 f o r a l l n$

$f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos \frac{n π x}{l} + b_{n} \sin \frac{n π x}{l})$ mm

Now,

$a_{0} = \frac{1}{l} \int_{- l}^{l} f (x) d x a_{0} = \frac{1}{1} [\int_{0}^{1} 0 d x + \int_{1}^{2} 1 d x] a_{0} = 1 . \int_{1}^{2} 1 d x a_{0} = x_{1}^{2} a_{0} = 2 - 1 = 1$

Step 3

hence, $a_{0} = \frac{1}{2}$

$b_{n} = \frac{1}{l} \int_{- l}^{l} f (x) \sin \frac{n πx}{l} d x b_{n} = \frac{1}{1} [\int_{0}^{1} 0 . \sin \frac{n πx}{1} d x + \int_{1}^{2} 1 . \sin \frac{n πx}{1} d x] b_{n} = \int_{1}^{2} 1 . \sin \frac{n πx}{1} d x i n t e g r a l o f \int \sin x d x = - \cos x b_{n} = - \cos {\frac{n πx}{n π}}_{1}^{2} b_{n} = - \frac{1}{n π} [\cos 2 n π - \cos n π] f o r a l l n \geq 1 b_{1} = - \frac{1}{π} [\cos 2 π - \cos π] = \frac{- 2}{π} b_{2} = - \frac{1}{2 π} [\cos 4 π - \cos 2 π] = 0 b_{3} = - \frac{1}{3 π} [\cos 6 π - \cos 3 π] = \frac{- 2}{3 π} b_{4} = - \frac{1}{4 π} [\cos 8 π - \cos 4 π] = 0 b_{5} = - \frac{1}{5 π} [\cos 10 π - \cos 5 π] = \frac{- 2}{5 π}$

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 1 images

Knowledge Booster

$Fourier Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions