We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Justify that there exists a real $A > 0$ such that for $x > A , q ( x ) > c ^ { 2 }$ ($q$ defined in III.A.2).