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





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




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