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)/(x)=(dα/ds·ds/dx)/(x)=(dα/ds)/(x)
2
cos
2
cos
3
cos
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))(α)
3
cos
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.)
P
1
x
1
y
1
,
y'
1
T
1
R
P
1
h
T
1
P
1
P
2
T
i