Question 1

The square norm of the function $f ( x ) = \cos ( 3 x )$ on the interval $[ 0 , \pi / 2 ]$ is

Accepted Answer

A)  1 
B)  $\pi$ 
C)  $\pi / 2$ 
D)  $\pi / 4$ 
E)  0 
A)  1 
B)  $\pi$ 
C)  $\pi / 2$ 
D)  $\pi / 4$ 
E)  0

Question 2

Consider the parameterized Bessel's differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( \alpha ^ { 2 } x ^ { 2 } - n ^ { 2 } \right) y = 0$ along with the conditions $y ( 0 )$ is bounded, $y ( 2 ) = 0$ . The solution of this eigenvalue problem is $\left( J _ { n } \left( z _ { n } \right) = 0 \right)$

Accepted Answer

A)  $\alpha = z _ { n } / 2 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$ 
B)  $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$ 
C)  $\alpha = z _ { n } , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$ 
D)  $\alpha = z _ { n } / 2 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$ 
E)  $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$ 
A)  $\alpha = z _ { n } / 2 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$ 
B)  $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$ 
C)  $\alpha = z _ { n } , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$ 
D)  $\alpha = z _ { n } / 2 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$ 
E)  $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$

Question 3

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

Accepted Answer

A)  $p$ , $q$ , $r$ are continuous on $[ a , b ]$ 
B)  $r ( x ) > 0$ and $p ( x )  0$ on $[ a , b ]$ 
D)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$ 
E)  $A _ { 1 } A _ { 2 } \neq 0$ 
A)  $p$ , $q$ , $r$ are continuous on $[ a , b ]$ 
B)  $r ( x ) > 0$ and $p ( x )  0$ on $[ a , b ]$ 
D)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$ 
E)  $A _ { 1 } A _ { 2 } \neq 0$

Question 4

The differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ is

Accepted Answer

A)  Legendre's equation 
B)  Bessel's equation 
C)  the Fourier-Bessel 
D)  the hypergeometric 
E)  none of the above 
A)  Legendre's equation 
B)  Bessel's equation 
C)  the Fourier-Bessel 
D)  the hypergeometric 
E)  none of the above

Question 5

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges at $x$ , to $( f ( x + ) + f ( x - ) ) / 2$ which of the following conditions need to be satisfied? Select all that apply.

Accepted Answer

A)  $f$ is continuous on $[ a , b ]$ 
B)  $f ^ { \prime }$ is continuous on $[ a , b ]$ 
C)  $f$ is piecewise continuous on $[ a , b ]$ 
D)  $f ^ { \prime }$ is piecewise continuous on $[ a , b ]$ 
E)  $f$ is integrable on $[ a , b ]$ 
A)  $f$ is continuous on $[ a , b ]$ 
B)  $f ^ { \prime }$ is continuous on $[ a , b ]$ 
C)  $f$ is piecewise continuous on $[ a , b ]$ 
D)  $f ^ { \prime }$ is piecewise continuous on $[ a , b ]$ 
E)  $f$ is integrable on $[ a , b ]$

Question 6

The square norm of the function $f ( x ) = 1 - x$ on the interval $[ 0,2 ]$ is

Accepted Answer

A)  2/3 
B)  $\sqrt { 2 / 3 }$ 
C)  1/3 
D)  $\sqrt { 1 / 3 }$ 
E)  0 
A)  2/3 
B)  $\sqrt { 2 / 3 }$ 
C)  1/3 
D)  $\sqrt { 1 / 3 }$ 
E)  0

Question 7

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ is

Accepted Answer

A)  $\lambda = n \pi , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$ 
B)  $\lambda = n \pi , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$ 
D)  $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$ 
E)  $\lambda = n \pi , y = \cos ( n \pi x ) + \sin ( n \pi x ) , n = 1,2,3 , \ldots$ 
A)  $\lambda = n \pi , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$ 
B)  $\lambda = n \pi , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$ 
D)  $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$ 
E)  $\lambda = n \pi , y = \cos ( n \pi x ) + \sin ( n \pi x ) , n = 1,2,3 , \ldots$

Question 8

The Fourier series of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ is

Accepted Answer

A)  $\sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right) + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
B)  $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
C)  $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
D)  $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
E)  $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
A)  $\sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right) + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
B)  $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
C)  $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
D)  $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$ 
E)  $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$

Question 9

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )$ , where

Accepted Answer

A)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$ 
B)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x$ 
C)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$ 
D)  $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$ 
E)  $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$ 
A)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$ 
B)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x$ 
C)  $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$ 
D)  $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$ 
E)  $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$

Question 10

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

Accepted Answer

A)  $r = 1 / ( x - a )$ are continuous on $[ a , b ]$ 
B)  $p ( x ) = x - a$ on $[ a , b ]$ 
C)  $q ( x ) = 0$ on $[ a , b ]$ 
D)  $A _ { 1 } A _ { 2 } = 0$ 
E)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$ 
A)  $r = 1 / ( x - a )$ are continuous on $[ a , b ]$ 
B)  $p ( x ) = x - a$ on $[ a , b ]$ 
C)  $q ( x ) = 0$ on $[ a , b ]$ 
D)  $A _ { 1 } A _ { 2 } = 0$ 
E)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$

Question 11

Consider the differential equation $y ^ { \prime \prime } + \lambda y = 0$ . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

Accepted Answer

A)  $y ( 0 ) = 0 , y ( 1 ) = 0$ 
B)  $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
C)  $y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0$ 
D)  $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$ 
E)  $y ( 0 ) + y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
A)  $y ( 0 ) = 0 , y ( 1 ) = 0$ 
B)  $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
C)  $y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0$ 
D)  $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$ 
E)  $y ( 0 ) + y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$

Question 12

Using the eigenfunctions of the previous problem, written as $g _ { n } ( x )$ , the Fourier-Bessel series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )$ , where

Accepted Answer

A)  $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$ 
B)  $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$ 
C)  $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$ 
D)  $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$ 
E)  $c _ { n } = \int _ { 0 } ^ { 2 } x ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$ 
A)  $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$ 
B)  $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$ 
C)  $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$ 
D)  $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$ 
E)  $c _ { n } = \int _ { 0 } ^ { 2 } x ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$

Question 13

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is not a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

Accepted Answer

A)  $r = 1 / ( x - a )$ on $( a , b )$ 
B)  $q ( x ) = 0$ on $[ a , b ]$ 
C)  $p ( x ) = x - a$ on $[ a , b ]$ 
D)  $A _ { 1 } A _ { 2 } = 0$ 
E)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$ 
A)  $r = 1 / ( x - a )$ on $( a , b )$ 
B)  $q ( x ) = 0$ on $[ a , b ]$ 
C)  $p ( x ) = x - a$ on $[ a , b ]$ 
D)  $A _ { 1 } A _ { 2 } = 0$ 
E)  $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$

Question 14

The solution of the eigenvalue problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ where $y$ is bounded on $[ - 1,1 ]$ , is

Accepted Answer

A)  $\lambda = n , y = P _ { n } ( x ) , n = 1,2,3 ,$ 
B)  $\lambda = n - 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n + 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = n ^ { 2 } , y = P _ { n } ( x ) , n = 1,2,3 , .$ 
E)  $\lambda = n ( n + 1 ) , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$ 
A)  $\lambda = n , y = P _ { n } ( x ) , n = 1,2,3 ,$ 
B)  $\lambda = n - 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$ 
C)  $\lambda = n + 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$ 
D)  $\lambda = n ^ { 2 } , y = P _ { n } ( x ) , n = 1,2,3 , .$ 
E)  $\lambda = n ( n + 1 ) , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$

Question 15

The Fourier series of the function $f ( x ) = x$ on $[ - 1,1 ]$ is

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 2 \cos ( n \pi x ) / ( n \pi )$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \cos ( n \pi x ) / ( n \pi )$ 
C)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 \sin ( n \pi x ) / ( n \pi )$ 
D)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \sin ( n \pi x ) / ( n \pi )$ 
E)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \sin ( n \pi x ) / ( n \pi )$ 
A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 2 \cos ( n \pi x ) / ( n \pi )$ 
B)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \cos ( n \pi x ) / ( n \pi )$ 
C)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 \sin ( n \pi x ) / ( n \pi )$ 
D)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \sin ( n \pi x ) / ( n \pi )$ 
E)  $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \sin ( n \pi x ) / ( n \pi )$

Question 16

Which of the following differential equations are in self-adjoint form? Select all that apply.

Accepted Answer

A)  $r ( x ) y ^ { \prime \prime } + r ^ { \prime } ( x ) y ^ { \prime } + \lambda y = 0$ 
B)  $y ^ { \prime \prime } + y ^ { \prime } + \lambda y = 0$ 
C)  $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ 
D)  $y ^ { \prime \prime } + \lambda y = 0$ 
E)  $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0$ 
A)  $r ( x ) y ^ { \prime \prime } + r ^ { \prime } ( x ) y ^ { \prime } + \lambda y = 0$ 
B)  $y ^ { \prime \prime } + y ^ { \prime } + \lambda y = 0$ 
C)  $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ 
D)  $y ^ { \prime \prime } + \lambda y = 0$ 
E)  $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0$

Question 17

The square norm of the function $f ( x ) = x ^ { 2 }$ on the interval $[ 0,1 ]$ is

Accepted Answer

A)  1/2 
B)  1/3 
C)  1/5 
D)  1 
E)  0 
A)  1/2 
B)  1/3 
C)  1/5 
D)  1 
E)  0

Question 18

The Fourier series of an odd function might Select all that apply.

Accepted Answer

A)  contain sine terms 
B)  contain cosine terms 
C)  contain a constant term 
D)  contain sine and cosine terms 
E)  contain sine, cosine, and constant terms 
A)  contain sine terms 
B)  contain cosine terms 
C)  contain a constant term 
D)  contain sine and cosine terms 
E)  contain sine, cosine, and constant terms

Question 19

The function $f ( x ) = | x |$ is Select all that apply.

Accepted Answer

A)  odd 
B)  even 
C)  neither even nor odd 
D)  continuous on $[ - \pi , \pi ]$ 
E)  discontinuous on $[ - \pi , \pi ]$ 
A)  odd 
B)  even 
C)  neither even nor odd 
D)  continuous on $[ - \pi , \pi ]$ 
E)  discontinuous on $[ - \pi , \pi ]$

Question 20

An example of a regular Sturm-Liouville problem is $y ^ { \prime \prime } + \lambda y = 0$ with boundary conditions Select all that apply.

Accepted Answer

A)  $y ( 0 ) + y ^ { \prime } ( 1 ) = 0 , y ( 1 ) = 0$ 
B)  $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
C)  $y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
D)  $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$ 
E)  $y$ is bounded on $[ - 1,1 ]$ 
A)  $y ( 0 ) + y ^ { \prime } ( 1 ) = 0 , y ( 1 ) = 0$ 
B)  $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
C)  $y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$ 
D)  $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$ 
E)  $y$ is bounded on $[ - 1,1 ]$

The square norm of the function $f ( x ) = \cos ( 3 x )$ on the interval $[ 0 , \pi / 2 ]$ is

The differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ is

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges at $x$ , to $( f ( x + ) + f ( x - ) ) / 2$ which of the following conditions need to be satisfied? Select all that apply.

The square norm of the function $f ( x ) = 1 - x$ on the interval $[ 0,2 ]$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ is

The Fourier series of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ is

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )$ , where

Consider the differential equation $y ^ { \prime \prime } + \lambda y = 0$ . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

Using the eigenfunctions of the previous problem, written as $g _ { n } ( x )$ , the Fourier-Bessel series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )$ , where

The solution of the eigenvalue problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ where $y$ is bounded on $[ - 1,1 ]$ , is

The Fourier series of the function $f ( x ) = x$ on $[ - 1,1 ]$ is

Which of the following differential equations are in self-adjoint form? Select all that apply.

The square norm of the function $f ( x ) = x ^ { 2 }$ on the interval $[ 0,1 ]$ is

The Fourier series of an odd function might Select all that apply.

The function $f ( x ) = | x |$ is Select all that apply.

An example of a regular Sturm-Liouville problem is $y ^ { \prime \prime } + \lambda y = 0$ with boundary conditions Select all that apply.

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 11: Orthogonal Functions and Fourier Series

The square norm of the function f(x)=cos⁡(3x)f ( x ) = \cos ( 3 x )f(x)=cos(3x) on the interval [0,π/2][ 0 , \pi / 2 ][0,π/2] is

The differential equation (1−x2)y′′−2xy′+λy=0\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0(1−x2)y′′−2xy′+λy=0 is

In order to be assured by a theorem that the Fourier Series of fff on [a,b][ a , b ][a,b] converges at xxx , to (f(x+)+f(x−))/2( f ( x + ) + f ( x - ) ) / 2(f(x+)+f(x−))/2 which of the following conditions need to be satisfied? Select all that apply.

The square norm of the function f(x)=1−xf ( x ) = 1 - xf(x)=1−x on the interval [0,2][ 0,2 ][0,2] is

The solution of the eigenvalue problem y′′+λy=0,y(0)=0,y(1)=0y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0y′′+λy=0,y(0)=0,y(1)=0 is

The Fourier series of the function f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 on [−1,1][ - 1,1 ][−1,1] is

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function f(x)f ( x )f(x) is ∑n=1∞cnPn(x)\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )∑n=1∞​cn​Pn​(x) , where

Consider the differential equation y′′+λy=0y ^ { \prime \prime } + \lambda y = 0y′′+λy=0 . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

Using the eigenfunctions of the previous problem, written as gn(x)g _ { n } ( x )gn​(x) , the Fourier-Bessel series for the function f(x)f ( x )f(x) is ∑n=1∞cngn(x)\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )∑n=1∞​cn​gn​(x) , where

The solution of the eigenvalue problem (1−x2)y′′−2xy′+λy=0\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0(1−x2)y′′−2xy′+λy=0 where yyy is bounded on [−1,1][ - 1,1 ][−1,1] , is

The Fourier series of the function f(x)=xf ( x ) = xf(x)=x on [−1,1][ - 1,1 ][−1,1] is

Which of the following differential equations are in self-adjoint form? Select all that apply.

The square norm of the function f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 on the interval [0,1][ 0,1 ][0,1] is

The Fourier series of an odd function might Select all that apply.

The function f(x)=∣x∣f ( x ) = | x |f(x)=∣x∣ is Select all that apply.

An example of a regular Sturm-Liouville problem is y′′+λy=0y ^ { \prime \prime } + \lambda y = 0y′′+λy=0 with boundary conditions Select all that apply.

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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The square norm of the function $f ( x ) = \cos ( 3 x )$ on the interval $[ 0 , \pi / 2 ]$ is

The differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ is

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges at $x$ , to $( f ( x + ) + f ( x - ) ) / 2$ which of the following conditions need to be satisfied? Select all that apply.

The square norm of the function $f ( x ) = 1 - x$ on the interval $[ 0,2 ]$ is

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ is

The Fourier series of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ is

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )$ , where

Consider the differential equation $y ^ { \prime \prime } + \lambda y = 0$ . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

Using the eigenfunctions of the previous problem, written as $g _ { n } ( x )$ , the Fourier-Bessel series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )$ , where

The solution of the eigenvalue problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ where $y$ is bounded on $[ - 1,1 ]$ , is

The Fourier series of the function $f ( x ) = x$ on $[ - 1,1 ]$ is

The square norm of the function $f ( x ) = x ^ { 2 }$ on the interval $[ 0,1 ]$ is

The function $f ( x ) = | x |$ is Select all that apply.

An example of a regular Sturm-Liouville problem is $y ^ { \prime \prime } + \lambda y = 0$ with boundary conditions Select all that apply.