Deck 6: Series Solutions of Linear Equations

Question 1

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

A) $( k + 2 ) ( k + 1 ) c _ { k + 2 } + c _ { k } = 0$
B) $( k + 2 ) ( k + 1 ) c _ { k } + c _ { k - 2 } = 0$
C) $( k + 1 ) k c _ { k + 2 } + c _ { k } = 0$
D) $( k + 1 ) k c _ { k } + c _ { k - 2 } = 0$
E) $( k - 2 ) ( k - 1 ) c _ { k - 2 } + c _ { k } = 0$

Answer 1

$( k + 2 ) ( k + 1 ) c _ { k + 2 } + c _ { k } = 0$

Question 2

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

A) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k + 1 } / ( 2 k + 1 ) !$
B) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k + 1 } / ( 2 k + 1 )$
C) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }$
D) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k - 1 } / ( 2 k - 1 ) !$
E) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k - 1 } / ( 2 k - 1 )$

Answer 2

$y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } ( - 1 ) ^ { k } x ^ { 2 k + 1 } / ( 2 k + 1 ) !$

Question 3

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 4$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Answer 3

an irregular singular point

Question 4

The solution of the previous problem is

A) $y = c _ { 1 } P _ { 1 / s } ( x ) + c _ { 2 } P _ { - 1 / 5 } ( x )$
B) $y = c _ { 1 } P _ { 5 } ( x ) + c _ { 2 } P _ { - 5 } ( x )$
C) $y = c _ { 1 } J _ { 5 } ( x ) + c _ { 2 } Y _ { 5 } ( x )$
D) $y = c _ { 1 } J _ { 1 / s } ( x ) + c _ { 2 } ^ { J } J _ { - 1 / 5 } ( x )$
E) $y = c _ { 1 } J _ { 1 / 25 } ( x ) + c _ { 2 } Y _ { 1 / 25 } ( x )$

Question 5

The interval of convergence of the power series in the previous problem is

A) $\{ 0 \}$
B) $( - 1,1 )$
C) $[ - 1,1 ]$
D) $( - 1,1 ]$
E) $( - \infty , \infty )$

Question 6

Consider the differential equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ . The indicial equation is $r ( r - 1 ) = 0$ . The recurrence relation is $c _ { k + 1 } ( k + r + 1 ) + ( k + r ) - c _ { k } ( k + r - 1 ) = 0$ . A series solution corresponding to the indicial root $r = 0$ is

A) $y _ { 1 } = x$
B) $y _ { 1 } = x ^ { 2 }$
C) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]$
D) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( 2 k - 3 ) ! ]$
E) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]$

Question 7

In the previous problem, a second solution is

A) $y _ { 2 } = e ^ { x }$
B) $y _ { 2 } = x \int e ^ { x } / x ^ { 2 } d x$
C) $y = 1 + \sum _ { k - 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = ( k - 1 ) / ( k ( k + 1 ) )$
D) $y = 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } , \text { where } c _ { k } = 1 / k ^ { 2 }$
E) none of the above

Question 8

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Question 9

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 12 y = 0$ is

A) Bessel's equation of order 12
B) Bessel's equation of order 3
C) Legendre's equation of order 12
D) Legendre's equation of order 3
E) Legendre's equation of order 4

Question 10

For the equation $\left( x ^ { 2 } - 16 \right) ^ { 3 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 1$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Question 11

Find three positive values of $\lambda$ for which the differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ has polynomial solutions.

A) 2, 6, 12
B) 1, 2, 3
C) 1, 4, 9
D) 2, 4, 6
E) 2, 6, 10

Question 12

The solution of the recurrence relation in the previous problem is

A) $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 )$
B) $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) ^ { 2 }$
C) $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 1 ) !$
D) $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k + 3 ) !$
E) $c _ { 2 k } = c _ { 0 } ( - 1 ) ^ { k } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } ( - 1 ) ^ { k } / ( 2 k ) !$

Question 13

The solution of the previous problem is

A) $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } P _ { - 3 } ( x )$
B) $y = c _ { 1 } P _ { 3 } ( x ) + c _ { 2 } Q _ { 3 } ( x )$ , where $Q _ { 3 } ( x )$ is given by an infinite series
C) $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$
D) $y = c _ { 1 } J _ { 3 } ( x ) + c _ { 2 } Y _ { 3 } ( x )$
E) $y = c _ { 1 } J _ { 12 } ( x ) + c _ { 2 } Y _ { 12 } ( x )$

Question 14

The singular points of the differential equation $x y ^ { \prime \prime } + y ^ { \prime } + y ( x + 2 ) / ( x - 4 ) = 0$ are

A) none
B) 0
C) 0, $- 2$
D) 0, 4
E) 0, $- 2$ , 4

Question 15

The recurrence relation for the differential equation $x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0$ is

A) $c _ { k } ( k + r ) ( k + r - 1 ) + c _ { k - 2 } = 0$
B) $c _ { k } ( k + r ) ( k + r - 1 ) - c _ { k - 2 } = 0$
C) $c _ { k } ( k + r + 1 ) ^ { 2 } - c _ { k - 2 } = 0$
D) $c _ { k } ( k + r + 2 ) ( k + r + 1 ) + c _ { k - 2 } = 0$
E) $c _ { k } ( k + r ) ( k + r + 1 ) - c _ { k - 2 } = 0$

Question 16

The radius of convergence of the power series $\sum _ { n = 1 } ^ { \infty } x ^ { n } / n !$ is

A) 0
B) 1
C) 2
D) $\infty$
E) none of the above

Question 17

The first four nonzero terms in the power series expansion of the function $f ( x ) = \sin x$ about $x = 0$ are

A) $1 - x + x ^ { 2 } / 2 - x ^ { 3 } / 3$
B) $x - x ^ { 3 } / 6 + x ^ { 5 } / 120 - x ^ { 7 } / 5040$
C) $x + x ^ { 3 } + x ^ { 5 } + x ^ { 7 }$
D) $1 + x ^ { 2 } / 2 + x ^ { 4 } / 4 + x ^ { 6 } / 6$
E) $1 - x ^ { 2 } / 2 + x ^ { 4 } / 24 - x ^ { 6 } / 720$

Question 18

The differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 25 \right) y = 0$ is

A) Bessel's equation of order $n$
B) Bessel's equation of order 1/25
C) Bessel's equation of order 1/5
D) Legendre's equation of order 1/25
E) Legendre's equation of order 1/5

Question 19

The radius of convergence of the power series solution of $y ^ { \prime \prime } + y = 0$ about $x = 0$ is

A) 0
B) 1
C) 2
D) $\infty$
E) none of the above

Question 20

The indicial equation for the differential equation $x y ^ { \prime \prime } + 2 y ^ { \prime } - x y = 0$ is

A) $r ( r - 1 ) = 0$
B) $r ( r + 2 ) = 0$
C) $r ( 2 r + 1 ) = 0$
D) $r ( 2 r - 1 ) = 0$
E) $r ( r + 1 ) = 0$

Question 21

The differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - 1 / 16 \right) y = 0$ is

A) Bessel's equation of order $n$
B) Bessel's equation of order 1/16
C) Bessel's equation of order 1/4
D) Legendre's equation of order 1/16
E) Legendre's equation of order 1/4

Question 22

The solution of the previous problem is

A) $y = c _ { 1 } P _ { 1 / 4 } ( x ) + c _ { 2 } P _ { - 1 / 4 } ( x )$
B) $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } P _ { 4 } ( x )$
C) $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$
D) $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$
E) $y = c _ { 1 } J _ { 1 / 16 } ( x ) + c _ { 2 } J _ { - 1 / 16 } ( x )$

Question 23

The radius of convergence of the power series $\sum _ { n = 1 } ^ { \infty } x ^ { n } / n$ is

A) 0
B) 1
C) 2
D) $\infty$
E) none of the above

Question 24

The singular points of $x ^ { 2 } ( x - 1 ) y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ are $x =$ Select all that apply.

A) 2
B) $- 1$
C) 0
D) 1
E) none of the above

Question 25

The recurrence relation for the differential equation $2 x y ^ { \prime \prime } - y ^ { t } + 2 y = 0$ is

A) $c _ { k + 1 } ( k + r ) ( 2 k + 2 r - 1 ) + 2 c _ { k } = 0$
B) $c _ { k + 1 } ( k + r ) ( k + r - 1 ) + 2 c _ { k } = 0$
C) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r - 1 ) - 2 c _ { k } = 0$
D) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r - 1 ) + 2 c _ { k } = 0$
E) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r ) + 2 c _ { k } = 0$

Question 26

Consider the differential equation $2 x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + ( 2 x - 1 ) y = 0$ The indicial equation is $2 r ^ { 2 } + r - 1 = 0$ . The recurrence relation is $c _ { k } [ 2 ( k + r ) + ( k + r - 1 ) + 3 ( k + r ) - 1 ] + 2 c _ { k - 1 } = 0$ . A series solution corresponding to the indicial root $r = - 1$ is $y = x ^ { - 1 } \left[ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right]$ , where

A) $c _ { k } = ( - 2 ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 3 ) ]$
B) $c _ { k } = - 2 ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]$
C) $c _ { k } = ( - 2 ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]$
D) $c _ { k } = ( - 2 ) ^ { k } / [ k ! ( - 1 ) ( 2 k - 3 ) ! ]$
E) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 1 \cdot 3 \cdots ( 2 k - 5 ) ]$

Question 27

The interval of convergence of the power series in the previous problem is

A) $\{ 0 \}$
B) $( - 1,1 )$
C) $[ - 1,1 ]$
D) $[ - 1,1 )$
E) $( - \infty , \infty )$

Question 28

The singular points of the differential equation $y ^ { \prime \prime } + y ^ { \prime } / x + y ( x - 2 ) / ( x - 3 ) = 0$ are

A) none
B) 0
C) 0, 2
D) 0, 3
E) 0, 2, 3

Question 29

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 0$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Question 30

The recurrence relation for the power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is (for $k = 0,1,2 , \ldots$ )

A) $( k + 2 ) ( k + 1 ) c _ { k + 2 } = c _ { k }$
B) $( k + 2 ) ( k + 1 ) c _ { k } = c _ { k - 2 }$
C) $( k + 1 ) k c _ { k + 2 } = c _ { k }$
D) $( k + 1 ) k c _ { k } = c _ { k - 2 }$
E) $( k - 2 ) ( k - 1 ) c _ { k - 2 } = c _ { k }$

Question 31

In the previous problem, a series solution corresponding to the indicial root $r = 1 / 2$ is $y = x ^ { 1 / 2 } \left\{ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right\}$ , where

A) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 3 \cdot 5 \cdot 7 \cdots ( 2 k - 3 ) ]$
B) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 1 \cdot 3 \cdot 5 \cdots ( 2 k - 3 ) ]$
C) $c _ { k } = - 2 ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 1 ) ]$
D) $c _ { k } = ( - 2 ) ^ { k } / [ k ! ( 2 k + 3 ) ! ]$
E) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 3 ) ]$

Question 32

The radius of convergence of the power series solution of $y ^ { \prime \prime } - y = 0$ about $x = 0$ is

A) 0
B) 1
C) 2
D) $\infty$
E) none of the above

Question 33

The indicial equation for the differential equation $2 x y ^ { \prime \prime } - y ^ { t } + 2 y = 0$ is

A) $r ( 2 r - 1 ) = 0$
B) $r ( 2 r - 3 ) = 0$
C) $r ( 2 r - 2 ) = 0$
D) $r ( r - 3 ) = 0$
E) $r ( r - 2 ) = 0$

Question 34

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

A) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) !$
B) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 )$
C) $y = c _ { 0 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }$
D) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 ) !$
E) $y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 )$

Question 35

The solution of the previous problem is

A) $y = c _ { 1 } P _ { 20 } ( x ) + c _ { 2 } P _ { - 20 } ( x )$
B) $y = c _ { 1 } P _ { 4 } ( x ) + c _ { 2 } Q _ { 4 } ( x )$ , where $Q _ { 4 } ( x )$ is given by an infinite series
C) $y = c _ { 1 } J _ { 4 } ( x ) + c _ { 2 } Y _ { 4 } ( x )$
D) $y = c _ { 1 } J _ { 1 / 4 } ( x ) + c _ { 2 } J _ { - 1 / 4 } ( x )$
E) $y = c _ { 1 } J _ { 20 } ( x ) + c _ { 2 } Y _ { 20 } ( x )$

Question 36

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 2 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = 2$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Question 37

For the differential equation $\left( x ^ { 2 } - 4 \right) ^ { 3 } y ^ { \prime \prime } - 2 x y ^ { \prime } + y = 0$ , the point $x = - 2$ is

A) an ordinary point
B) a regular singular point
C) an irregular singular point
D) a special point
E) none of the above

Question 38

The first four terms in the power series expansion of the function $f ( x ) = e ^ { 2 x }$ about $x = 0$ are

A) $1 + x + x ^ { 2 } + x ^ { 3 }$
B) $1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 }$
C) $1 + 2 x + 2 x ^ { 2 } + 4 x ^ { 3 } / 3$
D) $1 + 2 x + 2 x ^ { 2 } + 2 x ^ { 3 } / 3$
E) $1 + 2 x + 4 x ^ { 2 } + 8 x ^ { 3 }$

Question 39

The solution of the recurrence relation in the previous problem is

A) $c _ { 2 k } = c _ { 0 } / ( 2 k ) , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 )$
B) $c _ { 2 k } = c _ { 0 } / ( 2 k ) ^ { 2 } , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) ^ { 2 }$
C) $c _ { 2 k } = c _ { 0 } / ( 2 k ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 1 ) !$
D) $c _ { 2 k } = c _ { 0 } / ( 2 k + 2 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k + 3 ) !$
E) $c _ { 2 k } = c _ { 0 } / ( 2 k - 1 ) ! , c _ { 2 k + 1 } = c _ { 1 } / ( 2 k ) !$

Question 40

The differential equation is $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + 20 y = 0$ is

A) Bessel's equation of order 20
B) Bessel's equation of order 4
C) Legendre's equation of order n
D) Legendre's equation of order 20
E) Legendre's equation of order 4