The solution of the initial-value problem $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.
A)
$y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n x ^ { 2 n - 1 } } { 2 ^ { 2 n - 1 } ( n ! ) ^ { 2 } }$
B)
$y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { n } } { 2 ^ { n } ( 2 n ! ) }$
C)
$y ( x ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$
D)
$y ( x ) = 2 ^ { x } + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( n ! ) ^ { 2 } }$
E)
$y ( x ) = x + \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

We recognize the given differential equation as the Bessel's equation of order 0. Thus, the solution is expected to be a Bessel function of order 0. The power series solution of the given differential equation can be written as 
$$y(x)=\sum_{n=0}^{\infty}a_nx^n.$$ 
Differentiating it twice, we obtain
$$y'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^{n},$$
and 
$$y''(x)=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^{n}.$$
Substituting the power series expansions of $y(x)$, $y'(x)$, and $y''(x)$ into the differential equation, we obtain
$$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^{n}+\sum_{n=0}^{\infty}a_nx^{n+1}+\sum_{n=0}^{\infty}a_nx^{n+2}=0.$$
Note that we have used $a_0=1$ and $a_1=0$ that follow from the initial conditions. Equating the coefficients of $x^n$, we obtain the relation $a_{n+2}=-\frac{a_n}{(n+1)(n+2)}$ for $n\geq 0.$ This recurrence relation can be used to obtain the power series coefficients. 
It can be verified that the power series $\sum_{n=0}^{\infty}( - 1 ) ^ { n } \frac { x ^ { 2 n } } { 2 ^ { 2 n } ( n ! ) ^ { 2 } }$, which satisfies the given recurrence relation, is exactly the power series for the Bessel function of order 0 (also denoted by $J_0(x)$). Hence, the solution of the initial-value problem is $y(x)=J_0(x)$, which agrees with choice C.

The Solution of the Initial-Value Problem x2y′′+xy′+x2y=0,y(0)=1,y′(0)=0x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0x2y′′+xy′+x2y=0,y(0)=1,y′(0)=0

The Solution of the Initial-Value Problem $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$