Classification of Differential Equations

Definition 10.4 (Ordinary Differential Equation). If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable, it is an Ordinary Differential Equation (ODE).

Definition 10.5 (Partial Differential Equation). An equation involving only partial derivatives of a function of two or more independent variables is a Partial Differential Equation (PDE). For instance, with $y$ the unknown function of the single variable $x$: $\dfrac{dy}{dx}+2y=e^{-x}$, $\dfrac{d^2y}{dx^2}-5\dfrac{dy}{dx}-y=0$, and $\dfrac{dx}{dt}+\dfrac{dy}{dt}=3x-4y$ are all ODEs; whereas $\dfrac{\partial u}{\partial y}=-\dfrac{\partial u}{\partial x}$, $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0$, and $\dfrac{\partial^2u}{\partial x^2}=\dfrac{\partial^2u}{\partial t^2}-\dfrac{\partial u}{\partial t}$ are PDEs. This chapter deals only with ODEs.

Ordinary differential equations further split into linear and nonlinear ones.

Definition 10.6 (Linear ODE). A general linear ODE of order $n$ can be written

where $a_0(x),\ldots,a_n(x)$ and $g(x)$ are any functions of $x$ (possibly zero, possibly constant). Three things characterise linearity: (1) there is no product of $y$ (or its derivatives) with each other; (2) no transcendental function (trig, log, …) is applied to $y$ or any of its derivatives; (3) $y$ and its derivatives never sit "inside" another function such as $y'$ or $e^{y'}$. The coefficients $a_0,\ldots,a_n$ and $g(x)$ may themselves be any functions of $x$ (constant, linear, nonlinear, zero, or nonzero) — only $y$ and its derivatives are restricted.

Definition 10.7 (Nonlinear ODE). Simply, any ODE that is not linear: if a coefficient of $y,y',y'',\ldots$ itself depends on $y$ or its derivatives, or a power such as $(y')^2$ appears, or a nonlinear function of $y$ or $y'$ (e.g. $\sin y$ or $e^{y'}$) appears, the equation is nonlinear. Examples of linear ODEs: $\dfrac{dy}{dx}=ax^3$, $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}+2y=0$, $\dfrac{dy}{dx}+p(x)y=q(x)$; whereas $y\dfrac{dy}{dx}+\sin x=0$ is nonlinear (product of $y$ and $y'$). $y''+xy'=xy^2+7$ is nonlinear (the $y^2$ term); $y''+y'=x$ is linear; $y^2+y'=x$ is nonlinear (the $y^2$ term); $y''=y\sin x$ is linear (the coefficient of $y$ is $-\sin x$, a function of $x$ only, not of $y$ or its derivatives).

Definition 10.8 (Homogeneous vs. non-homogeneous, linear-equation sense). In the general linear form above, if $g(x)=0$ the equation is called homogeneous; otherwise non-homogeneous. (This use of "homogeneous" describes the linear equation's right-hand side being zero — it is a different meaning from the "homogeneous differential equation" of §10.6.3, where the phrase instead describes $\dfrac{dy}{dx}$ being expressible as a function of $\dfrac{y}{x}$ alone; the two uses of the same word must not be confused.) …