First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > slope fields

### Differential Equations Alpha List

 Boundary Value Problems Cauchy-Euler Equations Chebyshev Equations Chemical Concentration Classify Differential Equations Constant Coefficients Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay First Order, Linear Fluids (Mixing) Fourier Series Inhomogeneous ODE's Integrating Factors (Exact) Integrating Factors (Linear) Laplace Transforms Linear Systems Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance Second Order, Linear Separation of Variables Shifting Theorems Slope Fields Solve Initial Value Problems Square Wave Substitution Undetermined Coefficients Unit Impulse Function Unit Step Function Variation of Parameters Wronskian

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Here is a great wikibooks page explaining slope fields in more detail: WikiBooks - Slope Fields

 Source: Wikipedia - Slope Field

Slope Fields

Alternate Name For Slope Fields - Direction Fields

Slope Fields are two dimensional representations of the slope of a differential equation. Here is an example of what a slope field looks like. The little lines all over the graph are what we call the slope field lines and they indicate the slope at those points. The three graphs are three possible solutions to the differential equation that was used to produce the slope field.

As you can probably tell from the graph, slope fields are best drawn by a computer. There are many online graphing sites out there. You can also use winplot to plot slope fields.

Here are couple of videos explaining slope fields and how to graph them. The first video is much better and more accurate than the second. So if you have time for only one, the first one is best to watch.

This first video is an indepth introduction to slope fields and a few other introductory topics. He does not actually solve any differential equations using regular techniques in this video. He spends time explaining the concept of differential equations, which is important to understand.

 MIT OCW - slope fields

Here is another video that shows how to plot a slope field by hand. It is not the best video in the world but it has some good techniques. Make sure you read the comments we've written about the video before accepting everything that he does in this video. That said, this video is included to give you another perspective.

 another perspective