$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$
$\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\Ai}{\mathrm{Ai}}$
Problem 1. Find three first terms of the decomposition at $z=0$ of the general solution of \begin{align} &u''-u'-zu=0,\tag{1a}\\ &u''-(1+z)u=0,\tag{1b}\\ &u''+(1+z^2)u=0.\tag{1c} \end{align}
Problem 2. Find three first terms of the decomposition at $z=0$ of two linearly independent solutions of \begin{align} &u''-\frac{1}{z}u=0,\tag{2a}\\ &u''+u'-\frac{1}{z}u=0,\tag{2b}\\ &u''-\frac{z+1}{z}u=0.\tag{2c} \end{align}
Problem 3. Find three first terms of the decomposition at $z=0$ of two linearly independent solutions of \begin{align} &u''+\frac{1}{z}u'-\frac{1}{z}u=0,\tag{3a}\\ &u''+\frac{1}{z}u'+\frac{1}{z}u=0.\tag{3b}\\ \end{align}
Problem 4. Find three first terms of the decomposition at $z=0$ of two linearly independent solutions of \begin{align} &u''+\frac{1}{z}u'-\frac{z+1}{z^2}u=0,\tag{4a}\\ &u''+\frac{1}{z}u'+\frac{z+1}{z^2}u=0.\tag{4b} \end{align}
Problem 5. Find three first terms of the decomposition at $z=0$ of three linearly independent solutions of \begin{align} &u'''-\frac{1}{z^2}u=0,\tag{5a}\\ &u'''-\frac{1}{z}u=0.\tag{5b} \end{align}
Problem 6. Find three first terms of the decomposition at $z=\infty0$ of two linearly independent solutions of \begin{align} &u''-\frac{1}{z^3}u=0,\tag{6a}\\ &u''-\frac{z+1}{z^4}u=0.\tag{6c} \end{align}
Problem 7. Find three first terms of the decomposition at $z=\infty$ of two linearly independent solutions of \begin{align} &u''+\frac{1}{z}u'-\frac{1}{z^3}u=0,\tag{7a}\\ &u''+\frac{1}{z}u'+\frac{1}{z^3}u=0.\tag{7b}\\ \end{align}