Application. Let $\varepsilon > 0$. Let $\alpha \in \mathbb { R }$. Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function. We consider the following differential equation $$\left\{ \begin{array} { l }
u ^ { \prime \prime } ( t ) + u ( t ) = g \left( \frac { t } { \varepsilon } \right) \\
u ( 0 ) = \alpha , u ^ { \prime } ( 0 ) = 0
\end{array} \right.$$ (a) Justify the existence and uniqueness of a solution of (1), defined for $t \in \mathbb { R }$. (b) Calculate this solution using the method of variation of constants. We denote this solution $u _ { \varepsilon }$. (c) We assume that $g$ is $2 \pi$-periodic. Show that for all $t \in \mathbb { R } , u _ { \varepsilon } ( t )$ has a limit as $\epsilon \rightarrow 0 ^ { + }$, limit which one will calculate.
Application. Let $\varepsilon > 0$. Let $\alpha \in \mathbb { R }$. Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function. We consider the following differential equation
$$\left\{ \begin{array} { l }
u ^ { \prime \prime } ( t ) + u ( t ) = g \left( \frac { t } { \varepsilon } \right) \\
u ( 0 ) = \alpha , u ^ { \prime } ( 0 ) = 0
\end{array} \right.$$
(a) Justify the existence and uniqueness of a solution of (1), defined for $t \in \mathbb { R }$.\\
(b) Calculate this solution using the method of variation of constants. We denote this solution $u _ { \varepsilon }$.\\
(c) We assume that $g$ is $2 \pi$-periodic. Show that for all $t \in \mathbb { R } , u _ { \varepsilon } ( t )$ has a limit as $\epsilon \rightarrow 0 ^ { + }$, limit which one will calculate.