For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ Let $p \in \mathbb { R } ^ { * }$ and $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a sequence of real numbers. We assume that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence. Show that the function $f : x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $\left( E _ { p } \right)$ if and only if $$\left\{ \begin{array} { l } a _ { 0 } = 0 \\ n ( n + 1 ) a _ { n + 1 } = ( n - p ) a _ { n } , \quad \forall n \in \mathbb { N } ^ { * } \end{array} \right.$$
For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$
$$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$
Let $p \in \mathbb { R } ^ { * }$ and $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a sequence of real numbers. We assume that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence. Show that the function $f : x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $\left( E _ { p } \right)$ if and only if
$$\left\{ \begin{array} { l } a _ { 0 } = 0 \\ n ( n + 1 ) a _ { n + 1 } = ( n - p ) a _ { n } , \quad \forall n \in \mathbb { N } ^ { * } \end{array} \right.$$