Download Differential Equations – Definitions and Terminology

Differential Equations – Definitions and Terminology
An equation containing the derivatives of one or more dependent variables with respect to one or
more independent variables is said to be a differential equation.
Examples:
Differential equations are classified by:
1) Type
a. Ordinary Differential Equations
b. Partial Differential Equations
2) Order – The order of a differential equation is the order of the highest derivative in the
equation.
Note: Don’t confuse the order of a derivative with the power to which the derivative is raised.
3) Linearity
An nth-order ordinary differential equation is linear if and only if:
a) The dependent variable and all its derivatives are to the 1st degree.
b) Each coefficient depends at most on the independent variable x.
Examples of linear differential equations
Examples of nonlinear differential equations
Of course, what we will be concerned with in this course is finding solutions to differential equations.
To get you warmed up we will first let you verify a solution that has already been found.
dy
6 6
 20 y  24; y   e  20t
Example: #12 dt
5 5
Any function  , defined on an interval I and possessing at least n derivatives that are continuous on
I , which when substituted into an nth-order ordinary differential equation reduces the equation to an
identity, is said to be a solution of the equation on the interval.
The graph of a solution  if an ODE is called a solution curve.
Explicit Solution – A solution in which the dependent variable is expressed solely in terms of the
independent variable and constants. The solution given in #12 above was an explicit solution.
Implicit Solution – A relation or expression that defines a solution. ( y is not isolated)
Example: #16
Families of Solutions – When solving a 1st order differential equation F ( x, y, y )  0 we usually
obtain a solution containing a single arbitrary constant or parameter C. A solution containing an
arbitrary constant represents a set G ( x, y, C )  0 of solutions called a one-parameter family of
solutions. When solving an nth-order differential equation we seek an n-parameter family of
solutions.
Example: #23
A solution of a differential equation that is free of arbitrary parameters is called a particular solution.
Singular Solution – An “extra” solution that is not a member of a family of solutions.
Section 1.2 – Initial-Value Problems
Initial-value problems are differential equations that must be solved subject to prescribed side
conditions. That is, you will be given initial conditions that must be satisfied.
Example: #2
Existence and Uniqueness
The two fundamental questions that we will be dealing with most of the semester are:
1) Does a solution of the problem exist?
2) If a solution exists, is it unique?
Existence and Uniqueness Theorem for 1st Order Differential Equations
Let R be a rectangular region in the xy-plane defined by a  x  b, c  y  d that contains the point
x0 , y0  in its interior. If f ( x, y ) and f are continuous on R, then there exist some interval
y
I 0 : x0  h  x  x0  h, h  0 , contained in a  x  b , and a unique function y (x) , defined on I 0 , that is
dy
 f ( x, y ), y ( x0 )  y 0 .
a solution of the initial-value problem
dx
Examples: #8
#12
Read Section 1.3