Series Solution of Non-Linear Equation

 

The coefficients of the power series solutions of certain non-linear differential equations are generated by convolutions of the preceding coefficients.  One example is the differential equation

 

 

Among the solutions of this equation (with appropriate choices of a,b) are et, sin(t), cos(t), (A + Bt)n, A + Bt + Ct2, and . This last function represents the separation between any two objects in unaccelerated motion.  Other solutions include the cycloid relation for (non-rotating) gravitational free-fall, and the radial distance of a mass from a central point about which it revolves with constant angular velocity and radial freedom.

 

The power series solution of equation (1) can be written

 

 

where the coefficients ci satisfy the convolutions

 

 

with

 

Any choice of c0, c1, c2, and c3, with c1c2 not zero, determines the values of a,b and therefore all the remaining coefficients.  There are many interesting things about these sequences of ck values.  Focusing on just the sequences with |ck| = 1, k = 0,1,2,3, there are obviously 16 possible choices, but only 8 up to a simple sign change.  These 8 can be arranged as four groups of 2:

 

 

The coefficients in each group differ only in sign.  The coefficients in groups I and II diverge, and those in group IV are all units.  Only the group III sequences converge.  Interestingly, these coefficients are given very closely by

 

 

for k>2, where

 

 

Notice that the two possible values of w sum to 3.1415926...

 

The integer numerators and denominators of these ck sequences also have many interesting properties.  For example, primes p congruent to +1 (mod 4) first appear in the denominator at cp, whereas primes congruent to -1 (mod 4) first appear at .  The sequence of numerators is much less regular

 

 

Incidentally, the value of b in the ubiquitous equation (1) is essentially just a constant of integration, and the underlying relation is the derivative

 

 

where q = 3 for unaccelerated separations and q = 2 for (non-rotating) gravitational separations.  Isolating q and differentiating again leads to the basic relation, free of arbitrary constants,

 

 

Dividing by  gives the nice form

 

 

 

Return to MathPages Main Menu