Series Solution of NonLinear Equation 

The coefficients of the power series solutions of certain nonlinear 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 e^{t}, sin(t), cos(t), (A + Bt)^{n}, A + Bt + Ct^{2}, and _{}. This last function represents the separation between any two objects in unaccelerated motion. Other solutions include the cycloid relation for (nonrotating) gravitational freefall, 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 c_{i} satisfy the convolutions 

_{} 

with 
_{} 

Any choice of c_{0}, c_{1}, c_{2}, and c_{3}, with c_{1}c_{2} not zero, determines the values of a,b and therefore all the remaining coefficients. There are many interesting things about these sequences of c_{k} values. Focusing on just the sequences with c_{k} = 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 c_{k} sequences also have many interesting properties. For example, primes p congruent to +1 (mod 4) first appear in the denominator at c_{p}, 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 (nonrotating) gravitational separations. Isolating q and differentiating again leads to the basic relation, free of arbitrary constants, 

_{} 

Dividing by _{} gives the nice form 

_{} 

