Since I've been helped so many times on this forum, I wanted to "give back" something to the forum. I remember several months ago someone on this forum had a coupled system of differential equations to solve: an autonomous systems of ODEs that looked like it came from chemical kinetics, something like:
dx/dt = (5*x + y)/(2*x+3*y+5)
dy/dt = 2*x*y/(7*x+6*y+1)
Anyway, this summer I have been working on (obsessed with) an idea about rewriting practically any nonlinear differential equation in terms of purely functional equations. From these purely functional equations, one can derive LINEAR functional equations. The very preliminary sketches demonstrate that my ideas work for nonlinear PDEs as well.
The problem, of course, is that for even the SIMPLEST nonlinear ODE, I cannot crank out a fully computed example. I tried all summer, and used Maple to help determine some of the coefficients to see if I could see a pattern, but I had to put my idea aside.
Hence, I do not feel I have something complete enough for a published research paper. I will present the fragments I have at the Differential Algebra & Related Topics conference III in Newark, NJ in November 2008.
The general idea is this: rewrite one's system of ODEs above with all "indeterminate" powers and coefficient functions:
g(t)*x^a*y^b*(dx/dt)+h(t)*x^c*y^d*(dx/dt) = r(t)*x^u*y^v + s(t)*x^w*y^z
g2(t)*x^a2*y^b2*(dy/dt)+h2(t)*x^c2*y^d2*(dy/dt) = r2(t)*x^u2*y^v2 + s2(t)*x^w2*y^z2
Now, assume x = some functional form of g(t), h(t), r(t), s(t), g2(t), h2(t), r2(t), s2(t),
a, b, c, d, u, v, w, z, a2, b2, c2, d2, u2, w2, z2, such as
x= F(g,...,z2). Do the same for y, say, y=G(g,...z2).
Observe: there is nothing "unique" about these forms. In fact, they are not very well defined, since I am taking F of a functions, g(t), h(t), r(t) etc of the same variable, t. Hence, those functions are related. But, the trick is to pretend that they're not! (for the moment)
Multiply the first equation by x^alpha and the second equation by y^beta
for indeterminates alpha & beta. Rewrite the ODEs with NEW dependent variables x^(alpha+1) and y^(beta+1). One can then derive a purely functional relation that looks something like F(g(t),..a) = (F( (alpha+1)*g(t), ..., (a+alpha)/(1+alpha))^(1+alpha).
Next, take logarithms of both sides. Then, differentiate both sides, in turn, with respect to each of g(t), h(t), ...s2(t), a, b, c.. etc - ie w.r.t. EACH of the parameters in one's system of ODEs (again, pretending as if g, h, etc were independent variables).
From there, one can derive a LINEAR functional equation for EACH of these LOGARITHMIC DERIVATIVES.
Now, I have NOT gone farther than this for the general problem. This system of linear functional equations may not provide enough information to completely solve the problem. I have worked exclusively on the case dx/dt = g(t)*x^m + h(t)*x^n because there is a slightly DIFFERENT linear difference equation which arises in THIS particular case which I have not yet discovered for more complicated equations. However, I still did everything above for this "simple" equation that I explained above for the more complicated system in order to give more information about the form of the power series solution to this simpler case.