4.P. Problems to Chapter 4

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$

$\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\Ai}{\mathrm{Ai}}$

## Problems to Chapter 4

Problem 1. Find up to $O(\varepsilon^2)$ solution $u$ to \begin{align*} &u''-\varepsilon u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 2. Find up to $O(\varepsilon^2)$ solution $u$ to \begin{align*} &u''-\varepsilon \cos(x)u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 3. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon^2u''+u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 4. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon^2u''+u=\sin(x)&& 0< x< \pi\\ &u'(0)=u'(\pi)=0. \end{align*}

Problem 5. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon u''+ u'=\sin(x)&& 0< x< \pi\\ &u'(0)=u'(\pi)=1. \end{align*}

Problem 6. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon u''+ u'=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=1. \end{align*}