Differential Equation

An Equation in which the unknowns are functions. Solve for in the below.

Types

Ordinary Differential Equation Partial Differential Equation Linear Differential Equation Homogeneous Differential Equation Separable Differential Equation Systems of Differential Equations

Properties

Order

The order of a differential equation is the maximum derivative degree of the function of interest, e.g. , in the equation.

Techniques

Integrating Factor Separation of Variables

Special Examples

Population Dynamics

Linear Model

where is the net rate of change per capita

Nonlinear Logistic Model

Do not assume growth rate to be constant, but that it depends on Assume that when Assume that when Where is the intrinsic rate Where is the carrying capacity

Newton’s Law of Cooling

Well Stirred Solution

todo

Basketball

todo

Maximum Interval of Existence

The maximal interval of existence refewrs to the largest interval over which a solution to an initial value problem (IVP) of a differential equation is defined and unique.

Case 1

For both Linear Differential Equation and Nonlinear Differential Equation, if has discontinuities, solutions for may not exist beyond the discontinuities of , and the intervals of existence have become limited.

General Solution

(t,y(t))	Maximal Interval of Existence
(-6,2)
(11,10)
Basically, the given point determines , and depending on what is, the [[#maximum-interval-of-existence	Maximum Interval of Existence]] of the solution changes.

Case 2

For some Nonlinear Differential Equation, even if is differentiable everywhere, solutions for may not exist beyond some limited intervals.

The Maximum Interval of Existence follows the same pattern here. The point of case 2 is that even if your starting equation is continuous everywhere, your solution may not be, thus limiting the domain of