Consider the Parameterized Bessel's Differential Equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( \alpha ^ { 2 } x ^ { 2 } - n ^ { 2 } \right) y = 0$

Multiple Choice

Consider the parameterized Bessel's differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( \alpha ^ { 2 } x ^ { 2 } - n ^ { 2 } \right) y = 0$ along with the conditions $y ( 0 )$ is bounded, $y ( 2 ) = 0$ . The solution of this eigenvalue problem is $\left( J _ { n } \left( z _ { n } \right) = 0 \right)$

A) $\alpha = z _ { n } / 2 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$
B) $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$
C) $\alpha = z _ { n } , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$
D) $\alpha = z _ { n } / 2 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$
E) $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$