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Questions and Answers

Which type of differential equation is defined by a linear polynomial in the unknown function and its derivatives?

Quadratic equation

Linear equation

Ordinary differential equation (correct)

Partial differential equation

What is the term used to describe differential equations that may be with respect to more than one independent variable?

Quadratic equations

Ordinary differential equations

Partial differential equations (correct)

Linear equations

Which of the following is an example of a linear differential equation?

$y'' + 2xy' - 3y = 0$ (correct)

$y' + \sin(x)y = 0$

$y^2 + 3y' - 4x = 0$

$y''' + y'' - 2y' + y = 0$

In a linear differential equation, the functions $a_0(x), a_1(x), ..., a_n(x)$ and $b(x)$ can be any type of functions.

True Signup and view all the answers

What is the general form of a linear differential equation?

$a_0(x)y + a_1(x)y' + a_2(x)y'' + ... + a_n(x)y^{(n)} + b(x) = 0$ Signup and view all the answers

Which of the following statements about Fourier series is true?

Fourier series are a type of trigonometric series. Signup and view all the answers

Why are Fourier series useful in analyzing functions?

They simplify problems involving the function. Signup and view all the answers

What type of functions have Fourier series that converge to the original function?

Only smooth functions have Fourier series that converge. Signup and view all the answers

What did Joseph Fourier use Fourier series for?

To find solutions to the heat equation. Signup and view all the answers

Why can't Fourier series be used to approximate arbitrary functions?

Most functions have infinitely many terms in their Fourier series. Signup and view all the answers