Using power series methods, the solution of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ is
A) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / n ! \right]$
B) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n + 1 ) ) \right]$
C) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
D) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
E) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$

Question

Quizplus · Accepted Answer

The differential equation $x y'' - x y' + y = 0$ is a form of the Bessel equation, and its solution involves power series methods that typically result in terms involving $x$, $x \ln x$, and a power series. The correct option, E, correctly represents the structure of such a solution, including the $x \ln x$ term and a power series that correctly adjusts for the indices and factorial terms in the denominator, which are characteristic of solutions to differential equations of this type.

Using Power Series Methods, the Solution Of xy′′−xy′+y=0x y ^ { \prime \prime } - x y ^ { \prime } + y = 0xy′′−xy′+y=0

Using Power Series Methods, the Solution Of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$