# At a Glance - Differential Equations and Their Solutions

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.

' + " + 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

' + " + "' + x = 0

has order 3 because that's the highest derivative in the equation:

' + " + "' + 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.