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# Differential Equations

# Differential Equations Examples

#### Solutions to Differential Equations

Way back in algebra we learned that a solution to an equation is a value of the variable that makes the equation true. This is backwards kind of thinking we need for differential equations.To check...

#### Solving Differential Equations

Checking to see if a given function satisfies a given differential equation isn't too horrible of a task (at least for the functions you'll encounter in Calculus).The task of solving differential e...

#### Initial Value Problems

An Initial Value Problem (IVP) is a differential equation combined with one or more initial conditions. An initial condition gives some extra information about the solution. In order to be a soluti...

#### More About Solutions

Differential equations have two kinds of solutions: general and particular. The general solution to a differential equation is the collection of all solutions to that differential equation. A gener...

#### Word Problems

We can use differential equations to talk about things like how quickly a disease spreads, how fast a population grows, and how fast the temperature of cookies rises in an oven. Translating between...

#### Slope Fields and Solutions

If we have a slope field for a d.e. and a point in that slope field, we can sketch the solution that goes through that point. The little bits of tangent lines are like arrows telling the function w...

#### Equilibrium Solutions

An equilibrium solution is a solution to a d.e. whose derivative is zero everywhere. On a graph an equilibrium solution looks like a horizontal line.Given a slope field, we can find equilibrium sol...

#### Slopes (Again)

We know that the slope of a line is given byor bySince y is usually the dependent variable and x is usually the independent variable, you may also seeorThe symbol Δ is the Greek capital letter "De...

#### Tangent Line Approximations (Again)

Tangent line approximation can also be called local linearization, linear approximation, and probably a bunch of other names. The important thing is that you're using a line to approximate a curve....

#### The Scoop on Euler

Euler's Method is a bunch of tangent line approximations stuck together. The basic idea is that you start with a differential equation and a point. You do a tangent line approximation to get a new...

#### Accuracy and Usefulness of Euler's Method

Things to remember about Euler's Method:Euler's Method gives only approximate values unless the function happens to be a straight line, in which case Euler's Method gives exact values of the functi...