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

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.)


Contributed by: Izidor Hafner (February 2014)
Open content licensed under CC BY-NC-SA




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

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.