3. Fourier Trigonometric Series. -A
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Can you help me with question 3?
![**1. Boundary Value Problem with Nonhomogeneous ODE.**
For each choice of \( g(x) \) listed below, find all solutions to the following boundary value problem.
\[ 4y'' + \pi^2 y = g(x) \quad \text{for } 0 < x < 2, \quad y'(0) = 0, \, y'(2) = 0. \]
- (a) \( g(x) = 0 \)
- (b) \( g(x) = x \)
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**2. Eigenvalue Problem with First Derivative in Linear Operator.**
Find the eigenvalues and eigenfunctions for the boundary value problem,
\[ y'' + 4y' + \lambda y = 0 \quad \text{on } 0 < x < \pi, \quad y'(0) = 0, \, y'(\pi) = 0. \]
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**3. Fourier Trigonometric Series.**
Consider the function \( f(x) \) defined on \( (-\pi, \pi) \),
\[ f(x) = \begin{cases}
0, & -\pi < x < 0 \\
1, & 0 \leq x < \pi/2 \\
0, & \pi/2 \leq x < \pi
\end{cases} \]
and its Fourier series
\[ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right). \]
(a) Derive expressions for \( a_0, \, a_n \) and \( b_n \), for \( n = 1, 2, \ldots \).
(b) Write out the terms of the Fourier series through \( n = 5 \).
(c) Graph the periodic extension of \( f(x) \) on the interval \( (-3\pi, 3\pi) \) that represents the pointwise convergence of the Fourier series in part (b). At jump discontinuities, identify the value to which the series converges.
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**4. Eigenvalue Problem for Cauchy-Euler Equation.**
Find the eigenvalues and eigenfunctions for the boundary value problem,
\[ 4x^2y'' + 4xy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3143ece0-b2b2-417b-825f-ab4e468fd03e%2F00e1d1bf-e91f-4701-8b72-736d8c25e338%2Fehdigxz_processed.jpeg&w=3840&q=75)
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