$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Examples on linear and nonlinear differential equations
Linear ODE:
$$ u^{\prime} (t) = a(t)u(t) + b(t)$$
Nonlinear ODE:
$$ u^{\prime}(t) = u(t)(1 - u(t)) = u(t) - {\color{red}u(t)^2}$$
This (pendulum) ODE is also nonlinear:
$$ u^{\prime\prime} + \gamma\sin u = 0$$
because
$$ \sin u = u - \frac{1}{6}u^3 + \Oof{u^5},$$
contains products of \( u \)
