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

function

f

1

f

2

f

3

f

4

initial values for

x

0.

y

0.5

′

y

1.4

step

h

-0.09

number of steps

n

3

show arcs

show solution

plot / calculate

axes

This Demonstration shows a method of graphically approximating solutions of second-order differential equations. Let

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

be a given differential equation,

s

be the arc length of an integral curve, and

α

the angle between the tangent and the

x

axis. Therefore

y'=tan(α)

,

dx/ds=cos(α)

. Differentiate the first equation to get

y''=(dα/dx)/

2

cos

(x)=(dα/ds·ds/dx)/

2

cos

(x)=(dα/ds)/

3

cos

(x)

.

But

dα/ds=1/R

is the curvature where

R

is the radius of curvature. So rewrite the differential equation as

1/R=f(x,y,tan(α))

3

cos

(α)

, which gives the radius of curvature at a point on the curve as a function of the angle

α

.

From this, the following approximate construction is possible: at the initial point

P

1

(

x

1

,

y

1

)

,

construct the tangent of slope

y'

1

, and on the perpendicular line to it take a point

T

1

at distance

R

from

P

1

. Construct an arc of angle

h

with center at

T

1

and initial point

P

1

. The endpoint of the arc is

P

2

, which is the approximate second point on the curve. (The points

T

i

determine the evolute of the curve, approximately.)