To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 2 }$ on $\mathbb { R } _ { + } ^ { * }$ and that the function $U ( f )$ is a solution on $\mathbb { R } _ { + } ^ { * }$ of the differential equation $$y ^ { \prime \prime } - y ^ { \prime } = - \frac { f ( x ) } { x }$$
To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 2 }$ on $\mathbb { R } _ { + } ^ { * }$ and that the function $U ( f )$ is a solution on $\mathbb { R } _ { + } ^ { * }$ of the differential equation
$$y ^ { \prime \prime } - y ^ { \prime } = - \frac { f ( x ) } { x }$$