Section 2

index

Differential equations are equations containing derivatives, e.g.

$\frac{dy}{dx} = \sin(t^2)+1, \frac{d^2y}{dx^2} = y+t, \frac{\partial f(x,y)}{\partial x} = \sin(x+y)$

In this class, we would only care about the first two equations, where the third one contains more than one independent variable. All of these kinds of equations are named ODE (Ordinary Differential Equations).

The order of a differential equation is given by the order of the highest derivative in the equation.
We say an n-th order differential equation $F(t, y, y', y'', \ldots, y^{(n)}) = 0$ is linear if it can be written as

$a_n(t) y^{(n)} + a_{n-1}(t) y^{(n-1)} + \ldots + a_1(t) y' + a_0(t) y = g(t)$

Practice

ODE	Order	Linear/Nonlinear

Separable ODE

Separable ODE can be written as

$M(y) dy = N(t) dt$

and then take the integral for both sides. (This is only for first-order ODE, can be either linear or nonlinear.)

Integrating factors

A technique used to solve the 1st order linear ODE of the form

$\frac{dy}{dt} + p(t) y = q(t)$

(Only for first-order linear ODE)

Notation

Lagrange	Order	Newton	Leibniz
$y'$	1-st	$\dot{y}$	$\frac{dy}{dx}$
$y''$	2-nd	$\ddot{y}$	$\frac{d^2y}{dx^2}$
$y'''$	3-rd	$\dddot{y}$	$\frac{d^3y}{dx^3}$
$y^{(4)}$	4-th
$y^{(n)}$	n-th		$\frac{d^ny}{dx^n}$