Advertisement
Online Test Banks
Score higher
See Online Test Banks
eLearning
Learning anything is easy
Browse Online Courses
Mobile Apps
Learning on the go
Explore Mobile Apps
Dummies Store
Shop for books and more
Start Shopping

Solving Separable Differential Equations

Differential equations become harder to solve the more entangled they become. In certain cases, however, an equation that looks all tangled up is actually easy to tease apart. Equations of this kind are called separable equations (or autonomous equations), and they fit into the following form:

image0.png

Separable equations are relatively easy to solve. For example, suppose that you want to solve the following problem:

image1.png

You can think of the symbol

image2.png

as a fraction and isolate the x and y terms of this equation on opposite sides of the equal sign:

ey dy = sin x dx

Now integrate both sides:

image3.png

In an important sense, the previous step is questionable because the variable of integration is different on each side of the equal sign. You may think “No problem, it’s all integration!” But imagine if you tried to divide one side of an equation by 2 and the other by 3, and then laughed it off with “It’s all division!” Clearly, you’d have a problem. The good news, however, is that integrating both sides by different variables actually produces the correct answer.

C1 and C2 are both constants, so you can use the equation C = C2C1 to simplify the equation:

ey = –cos x + C

Next, use a natural log to undo the exponent, and then simplify:

ln e y = ln (–cos x + C)

y = ln (–cos x + C)

To check this solution, substitute this value for y into both sides of the original equation:

image4.png
  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
Advertisement
Advertisement

Inside Dummies.com

Dummies.com Sweepstakes

Win an iPad Mini. Enter to win now!