Question: solving a differential equation

Here is my code

> restart;

> with(plots);

> with(LinearAlgebra);

> nmax:=1:

> Order:=nmax+1:

> de1:=diff(F(eta,R),eta$4)+alpha*eta*diff(F(eta,R),eta$3)+2*alpha*diff(F(eta,R),eta$2)-R*(diff(F(eta,R),eta)*diff(F(eta,R),eta$2)-F(eta,R)*diff(F(eta,R),eta$3));

> S1:=series(eval(de1,F(eta,R)=add(F[n](eta)*R^n,n=0..nmax)),R):

> des1:=[seq(coeff(S1,R,k),k=0..nmax)]:

> bcs:=seq(seq(F[n,m](1) = `if`(n=0 and m=0, 1,  0), n=0..nmax), m=0..nmax),

          seq(seq(F[n,m](-1) = `if`(n=0 and m=0, -1,  0), n=0..nmax), m=0..nmax),

          seq(seq(D(F[n,m])(1) = 0, n=0..nmax), m=0..nmax),

          seq(seq(D(F[n,m])(-1) = 0, n=0..nmax), m=0..nmax);

> for k from 1 to nmax+1 do     



end do;

I have my differential equation. I am writing it in terms of a power series F=∑RFn from n=0 to nmax. Equating coefficients of powers of R. Then I’m writing the resulting equations from that in terms of a power series  Fn=∑αFnk  from k=0 to nmax for each n and equating coefficients of powers of alpha and trying to solve for each equation to get an equation for F which will use F=∑∑αRFnk .

I need to isolate each element  of the arrays des[k] and solve using the boundary conditions stated but I can’t figure out how. Once F[0,0] is found I can use it to find F[0,1] which can then be used to find F[0].

I am currently using nmax=1 so I can see what is going to but I want to increase this to about 20 or 30.

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