We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ : $$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$ Let $x _ { 0 }$ in $] 0 , + \infty [$. By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants. Deduce that there exist two real numbers $A$ and $B$ such that $$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$.
By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants.
Deduce that there exist two real numbers $A$ and $B$ such that
$$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$