Differential Equations

Derivative of functions with respect to independent variables

$a_nxy^{(n)} + a_{n-1}xy^{(n-1)} + \dots + \;a_2xy'' + a_1xy' + a_0xy=f(x)$

Order – the degree of the highest derivative
Ordinary – a function with only one variable
Partial – a function under multiple variables
ODE – "Ordinary Differential Equation"
Linear – *described later
Homogeneous *described later

First Order ODE

An Ordinary Differential Equation is an equation with derivative taken with respect to one variable.

The Picard-Lindelof Theorem

Linear ODE

A first order linear ODE generally follows the structure

$y' +P(x)y = Q(x)$

Where the degree of every derivative is to the first degree

An algebraic linear equation:

$ay_1+y_2=c$

A differential linear equation:

$a(x)y'+b(x)y=c(x)$

Therefore $y'+ x^2y=5$ is linear while $y' + y^2 = 0$ is not linear.

Homogeneity

A homogenous differential equation has the condition that every term in the differential equation has a variable taken to a derivative.

The following is homogeneous:

$y''+2y=0$

The following is non-homogeneous due to the $-5$:

$y'''+4x^7y''+sin(x)y-5=0$

Separable ODE

Whenever a Linear First Order Differential Equation is Homogeneous, the equation can be separated. An ODE expressed in the form

$\frac{dy}{dx} = g(x)y$

can be decomposed into

$\frac{1}{h(y)}=g(x)dx \quad \text{or} \quad \int \frac{1}{h(y)}=\int g(x)dx$

which reduces to

$ln(y)= H(x) + C \quad \text{or} \quad y = Ce^{H(x)}$

Example:

$\frac{dy}{dx} = 4y\sin(3x) \qquad (1)$

$\frac{1}{y}dy=4\sin(3x)dx \qquad (2)$

$\int\frac{1}{y}dy = \int 4\sin(3x)dx \qquad (3)$

$\ln(y) = -\frac{4}{3}\cos(3x)+C \qquad (4)$

$y= Ce^{-\frac{4}{3}\cos(3x)} \qquad (5)$