hi Judah
my answer is that
diff(x,t) is an arbitrary function,so we cannot do taylor expansion
with this function.
It is almost meaningless for all computer algebra system.
So your problem shoud be understood a spectral problem.
dx/dt=sum(b[i]*t^i,i,0,n);
f(x)=sum(a[i]*x^i,i,0,m);
so,
x=c+sum(b[j]/(j+1)*t^(j+1),j,0,n);
solve e[0],given a[i],b[i]
subst([x=...],f(x))=sum(e[i]*(dx/dt)^i,i,0,k);
after all with maxima
(C1) depends(x,t)$
(C2) gradef(x(t),sum(b[i]*t^i,i,0,3))$
(C3) f(x):=sum(a[i]*x^i,i,0,3)$
(C4) subst([x=c+sum(b[j]/(j+1)*t^(j+1),j,0,3)],f(x))$
(C5) sum(e[i]*sum(diff(x(t),t)^j,j,0,3),i,0,3)$
(C6) ppu:expand(D5-D4)$
(C7) eq1:coeff(ppu,t,0)$
(C8) eq2:coeff(ppu,t,1)$
(C9) eq3:coeff(ppu,t,2)$
(C10) solve([eq1,eq2,eq3],[e[0],e[1],e[2]]);
convergence is another problem,I think.
There is a nice book,we obtain pdf version from internet.
I recommend this book,he mainly use maple but we can use maxima.
Chebyshev and Fourier Spectral Methods
written by John P.Boyd
University of Michigan
thank
Gosei Furuya (go_furuya@infoseek.jp)