A differential equation (d.e.) is any equation that has one or more derivative in it. These can be first derivatives, second derivatives...whatever.
Sample Problem
The following are differential equations.
y ' + y " + xy = 0
Sample Problem
The following are not differential equations, because they don't contain any derivatives.
x2 + y2 = 8
x + xy – y + 9 = 0
x = 9
The order of a differential equation is the highest derivative that occurs in that differential equation.
Sample Problem
The differential equation
y ' + y " + y "' + x = 0
has order 3 because that's the highest derivative in the equation:
y ' + y " + y "' + x = 0.
Sample Problem
The differential equation
has order 1 because it only contains a first derivative.
A d.e. of order 1 is called a first-order differential equation, and a d.e. of order 2 is called a second-order differential equation. These are the kinds of differential equations that you'll probably see most often.
Exercise 1
Determine if the equation is a differential equation.
f (2)(x) + f (x) = 7x
Exercise 2
Determine if the equation is a differential equation.
Exercise 3
Determine if the equation is a differential equation.
x2 + f 2(x) = 0
Exercise 4
Determine if the equation is a differential equation.
x2 + y2 = 4xy
Exercise 5
Determine if the equation is a differential equation.
Exercise 6
Determine if the equation is a differential equation.
y + y2 + y3 = x
Exercise 7
Determine if the equation is a differential equation.
Exercise 8
Determine if the equation is a differential equation.
y' + 2y'' = 3 – x
Exercise 9
Determine the order of the differential equation.
Exercise 10
Determine the order of the differential equation.
y" + y' – y = 0
Exercise 11
Determine the order of the differential equation.
f '(x) = x2 + 3x + 5
Exercise 12
Determine the order of the differential equation.
f (x) – f 3(x) + f (2) = 7x
Exercise 13
Determine the order of the differential equation.
