Introduction to Differential Equations
Equations which are composed of an unknown function and its derivatives.
Types
Ordinary Differential Equations
When a differential equation involves one independent variable, and one or more dependent variables.
An example:
Partial Differential Equations
When a differential equation involves more than one independent variables, and more than one dependent variables.
Linear
A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function (dependant variable) and its derivatives, that is an equation of the form:
Here:
are all differentiable functions of , doesn’t depend on is the unknown function denotes the th derivative of
Nonlinear
Nonlinear differential equations are any equations that cannot be written in the above form. In particular, these include all equations that include:
and/or its derivatives raised to any power other than - nonlinear functions of
or any of its derivative - any product or function of these
Properties of Differential Equations
Order
Highest order derivative.
Degree
Power of highest order derivative.
Picard’s Existence and Uniqueness Theorem
Consider the below IVP.
Suppose:
If