Solving+Differential+Equations+by+Separating+Variables;+Exponential+Growth+and+Decay

== =**Solving Differential Equations by Separating Variables: Exponential Growth and Decay **=

To solve differential equations, many different methods can be used. Some are less accurate than others, but one of the first methods that should be tried when given a differential equation is Separating Variables. The basic premise is this: Put all x's on one side, all y's on the other, and integrate both sides, resulting (hopefully) in something that can be solved for y.

This works because the left side (with y in it) also has dy, and is integrated in terms of y. The right side (with x in it) also has dx, and is integrated in terms of x.

Just as one can add an integer to both sides, or multiply both sides of an equation by something, one can integrate both sides of an equation and keep it equivalent.

Separating Variables is often used to solve equations and get Exponential growth/decay equations, which often model real-world situations.

Example 1
For this equation, here is the slope field for reference (we will not be using it)

__Integrate both sides__ __Set a new C, C1 = __ __C1 will just be called C henceforth.__
 * __Solve it:__** Move all the y's to the left side, all the x's to the right.

__This is only a general solution... Say you wanted to solve using the original condition y(0) = 10__

__Particular solution:__ //can then be factored into://

Here is the graph of the solution (below).

Example 2
For this equation, here is the slope field for reference (we will not be using it)

__**Solve it:**__ Separate all the y's to the left, all the x's to the right __Integrate both sides__ __Set C1 =. C1 will just be called C henceforth.__

__Solve for y if y(0) = 5.__

__Particular Solution__

Here is a graph of the solution below. Notice that is fits the slope field (pretend the axes are perfect.)

Other Stuff
Graphs do not really seem necessary as this is process only involves mathematical operations, and we never have to calculate intersections, points of inflections, critical max or mins, etc.

As always, here is a video I shamelessly stole from the Internet.

Skip to about 4:00 for the first example. It is kind of slow-paced, it is an introduction to differential equations. media type="youtube" key="60VGKnYBpbg" width="425" height="350"
 * Useful Video (Taken from MIT OCW)**

Sources: Calculus, Concepts and Applications (Foerster) MIT OpenCourseWare Lectures WolframAlpha http://mathplotter.lawrenceville.org XKCD

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