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 $] 0 , + \infty [$. We set for $x > 0$ $$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$ a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$ $$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$ Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$. b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval. Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.
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 $] 0 , + \infty [$. We set for $x > 0$
$$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$
a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$
$$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$
Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$.
b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval.
Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.