For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that $(E_p)$ has non-identically zero polynomial solutions if and only if $p \in \mathbb { N } ^ { * }$. Show that then, the non-zero polynomial solutions of $(E_p)$ are of degree $p$ and belong to the vector space $E$.
For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$
$$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$
The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that $(E_p)$ has non-identically zero polynomial solutions if and only if $p \in \mathbb { N } ^ { * }$. Show that then, the non-zero polynomial solutions of $(E_p)$ are of degree $p$ and belong to the vector space $E$.