Study Guide

Differential Equations

Differential Equations Introduction

Samantha has finished her Ph.d. in meteorology, and has now joined forces with the local Shmoopville weather team. She loves getting in front of the green-screen and showing her weather chops. Sometimes she wears green, pretending to have an invisible torso. Beyond standing in front of the camera, Sam also likes to play around with the weather computers that are used to model the clouds we see on screen. These computer models use changes with respect to time (derivatives) in atmospheric variables like temperature and pressure to guess the weather. Snow in Florida? Tornado in Alaska? There's a model for that.

Differential equations are used to organize and figure out complicated problems. All of these derivative variables, along with some regular old variables, can be tossed into a differential equation. Because of differential equations, we can sit inside, eating cereal in our PJ's, and know exactly what to expect it to be like outside all day.

Differential Equations Resources


Differential Equations: Growth and Decay
Bread mold never looked so exciting. Learn how to describe the green growth on your slice of pumpernickel with a differential equation.

Initial Value Problems with Growth and Decay
A.K.A. I.V.P.’s with D.E.’s. Like this sentence, it sounds more complicated than it is. This link will decrypt these problems.

Differential Equations: Slope Fields
If you’re terrible at mini golf, learning how to model the greens using differential equation slope fields probably won’t help. But you can be good at modeling slope fields.


Introduction to Differential Equations
How long does it take to run to the fridge, make a sandwich, and sit back down to watch this video on position, velocity and acceleration? If you couldn’t answer before watching this video, now you can.

Verifying Solutions to Differential Equations
Math is a language, and its sentence is an equation. To make sure you chose a correct solution to a differential equation, plug it in like any other equation. This link shows you how.

Initial Value Solution to a Differential Equation
Without initial values, differential equations orbit in space with infinitely many solutions. Bring them back to Earth using initial values.

Games and Tools

Visual Differential Equation Solver
Some differential equation solutions are the most eye-catching beasts of functions you’ll see. Check out the terrifying Legendre function and the mystifying Bessel function.

Tool to Look at Initial Value Problems of Common Differential Equations

Fro-yo with math: create your own differential equation. Add sprinkles and other toppings with your own initial values. 

Visualize Slope Fields
If you’re still having trouble with your putting game, you can visualize the entire green here with this tool. Input your own differential equation and you’ll get the slope field around the origin.