Exam 6: Series Solutions of Linear Equations

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

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

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

The solution of the previous problem is

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.