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$$ \begin{align} L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{1}{C}q=\varepsilon_0\cos wt \\ ml\frac{d^2\theta}{dt^2}+cl\frac{d\theta}{dt}+mg\sin\theta=F_0\cos wt \\ \frac{\partial u}{\partial t}=D\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) \end{align} $$
全部都是 $2^{\text{nd}}$-order, 因為 independent variable $t$ 的 order 等於 2.
(1) and (2) 是 ordinary differential equation (ODE)
(3) 是 partial differential equation (PDE)
(1) and (3) 是 linear, (2) 是 nonlinear, 因為有 $\sin\theta$.
Linear N-order ODE/PDE 有解析解, 所以 “Differential Equations for Engineers” 這門課主要講這部分.
不過如果是 non-linear $n^\text{th}$-order ODE 雖沒有解析解, 但可利用 numerical methods 求近似解.
The general linear third-order ODE, where $y=y(x)$:
$$ a_3(x)y'''+a_2(x)y''+a_1(x)y'+a_0(x)y=b(x) $$
where the $a$ and $b$ coefficients can be any function of $x$.