9717

Smirnoff's Graphic Solution of a Second-Order Differential Equation

This Demonstration shows a method of graphically approximating solutions of second-order differential equations. Let be a given differential equation, be the arc length of an integral curve, and the angle between the tangent and the axis. Therefore , . Differentiate the first equation to get .
But is the curvature where is the radius of curvature. So rewrite the differential equation as , which gives the radius of curvature at a point on the curve as a function of the angle .
From this, the following approximate construction is possible: at the initial point construct the tangent of slope , and on the perpendicular line to it take a point at distance from . Construct an arc of angle with center at and initial point . The endpoint of the arc is , which is the approximate second point on the curve. (The points determine the evolute of the curve, approximately.)

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Reference
[1] V. I. Smirnoff, Lectures in Higher Mathematics, Vol. 2, Moscow: Nauka, 1967 pp. 48–50.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+