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 $x > 0$. Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$. Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.
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 $x > 0$.
Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.