We study the differential equation with unknown $y$ $$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$ We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$. Let $y$ be a solution in $E$ of (III.1). We set $z ( x ) = \sqrt { x } y ( x )$ for all $x \in ] 0 , + \infty [$. Show that $z$ is a solution in $E$ of a differential equation of the type $$z ^ { \prime \prime } + q z = 0 \tag{III.2}$$ with $q \in E$. Specify the expression of the function $q$ and verify that $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.
We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Let $y$ be a solution in $E$ of (III.1). We set $z ( x ) = \sqrt { x } y ( x )$ for all $x \in ] 0 , + \infty [$.
Show that $z$ is a solution in $E$ of a differential equation of the type
$$z ^ { \prime \prime } + q z = 0 \tag{III.2}$$
with $q \in E$. Specify the expression of the function $q$ and verify that $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.