- Topics At a Glance
**Differential Equations and Their Solutions**- Solutions to Differential Equations
- Solving Differential Equations
- Initial Value Problems
- More About Solutions
- Word Problems
- Slope Fields
- Slope Fields and Solutions
- Equilibrium Solutions
- Euler's Method
- Slopes (Again)
- Tangent Line Approximations (Again)
- The Scoop on Euler
- Accuracy and Usefulness of Euler's Method
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

A **differential equation (d.e.)** is any equation that has one or more derivative in it. These can be first derivatives, second derivatives...whatever.

The following are differential equations.

*y *' + *y *" + *xy* = 0

The following are not differential equations, because they don't contain any derivatives.

*x*^{2} + *y*^{2} = 8

*x* + *xy* – *y* + 9 = 0

*x* = 9

The **order** of a differential equation is the highest derivative that occurs in that differential equation.

The differential equation

d*y *' + *y *" + *y *"' + *x* = 0

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

*y *' + *y *" + *y *"' + *x* = 0.

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*) = 7*x*

Exercise 2

Determine if the equation is a differential equation.

Exercise 3

Determine if the equation is a differential equation.

*x*^{2} + *f* ^{2}(*x*) = 0

Exercise 4

Determine if the equation is a differential equation.

*x*^{2} + *y*^{2} = 4*xy*

Exercise 5

Determine if the equation is a differential equation.

Exercise 6

Determine if the equation is a differential equation.

*y* + *y*^{2} + *y*^{3} = *x*

Exercise 7

Determine if the equation is a differential equation.

Exercise 8

Determine if the equation is a differential equation.

*y*' + 2*y*'' = 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*) = *x*^{2} + 3*x* + 5

Exercise 12

Determine the order of the differential equation.

*f* (*x*) - *f* ^{3}(*x*) + *f* ^{(2)} = 7*x*

Exercise 13

Determine the order of the differential equation.