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

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 .

y''=f(x,y,y')

s

α

x

y'=tan(α)

dx/ds=cos(α)

y''=(dα/dx)/cos(x)=(dα/ds·ds/dx)/cos(x)=(dα/ds)/cos(x)

2

2

3

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 .

dα/ds=1/R

R

1/R=f(x,y,tan(α))cos(α)

3

α

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.)

Px,y

1

1

1

,

y'

1

T

1

R

P

1

h

T

1

P

1

P

2

T

i