The solution of the initial-value problem $x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ 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 differential equation as the Bessel equation of order 0. Thus, we seek a solution in the form of a power series: $y(x) = \sum\limits_{n=0}^\infty a_n x^n$. Taking derivatives and plugging into the differential equation, we get:

$$x^2y''+xy'+x^2y = \sum\limits_{n=0}^\infty n(n-1)a_nx^n + \sum\limits_{n=0}^\infty na_nx^n + \sum\limits_{n=0}^\infty a_nx^{n+2} = 0$$

Shifting the indices of the first sum, we combine the three sums and get:

$$\sum\limits_{n=2}^\infty [(n-1)(n-2)+n]a_nx^n + a_0 + a_1x = 0$$

Since this must hold for all values of $x$, we must have $a_0 + a_1x = 0$ and $(n-1)(n-2)a_n + na_n = 0$ for $n\geq 2$. Solving, we get $a_1 = -a_0$ and $a_n = \frac{(-1)^n}{2^{2n}(n!)^2}a_0$ for $n\geq 2$. Finally, using the initial conditions, we get $a_0 = 1$, so the solution is:

$$y(x) = a_0 \left(1 - \frac{x^2}{2^2\cdot 1^2} + \frac{x^4}{2^4\cdot 2^2\cdot 1^2} - \frac{x^6}{2^6\cdot 3^2\cdot 2^2\cdot 1^2} + \cdots \right) = \sum\limits_{n=0}^\infty \frac{(-1)^n}{2^{2n}(n!)^2}x^{2n}$$

which is the answer choice C.

The Solution of the Initial-Value Problem x2ytt+xy′+x2y=0,y(0)=1,yt(0)=0x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0x2ytt+xy′+x2y=0,y(0)=1,yt(0)=0

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