Classifying Differential Equations – Section 1.1
1. Classifying a differential equations as ordinary or partial:
Examples:
dy
 x is an example of an ordinary differential equation
dx
f f

 6 is an example of a partial differential equation
x y
Text examples: Page 4, examples (5) and (6).
2. Classifying a differential equation by order:
The order of an ordinary differential equation is the order of the highest-order
derivatives present in the equation. (text, page 4)
Examples:
dy
 x ; the order is 1.
dx
d3y
dy
2
 x ; the order is 3.
3
dx
dx
f f

 6 ; the order is 1.
x y
Text examples: page 4
3. Classifying an ordinary differential equation as linear or non- linear:
A ordinary DE is linear if:
a. the dependent variable and its various derivative occur to the first degree only
b. That no products of the dependent variable and/or its derivatives are present.
c. No transcendental functions (sine, cosine, ey , arcsin, log, etc.) of the dependent
variable and/or its derivatives are present.
Examples:
Linear
3
d y
dy
2
x
3
dx
dx
Nonlinear
3
d y
dy

y
 x;
dx 3
dx
d 3 y dy 2
( )  x
dx3
dx
d3y
 sin x
dx 3
d3y
 sin y
dx3
d 3x
t

e
dt 3
d 3x
x

e
dt 3
(text reference, page 5)