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 $a > A$, $z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, and $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
We denote $\left. I _ { a } = \right] a , a + \pi / c \left[ \right.$ and assume that $\varphi _ { n }$ has no zeros on $I _ { a }$.
Determine the signs of $W ( a ) , W ( a + \pi / c )$ and of $W ^ { \prime }$ on $I _ { a }$ and reach a contradiction. Deduce that $\varphi _ { n }$ has a zero in every interval $I _ { a }$ with $a > A$.
One may distinguish cases according to the sign of $\varphi _ { n }$ on $I _ { a }$.