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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...
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...
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...
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...
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