Expand in series an equation of two variables

Question

I am sorry if this has been asked, I don't really know how to formulate the search to find it.

We have some system of equations (that are equal to zero), for example (not sure how to enter expression with partial derivatives so it can be copy-pasted):

expr={4 (-1 + 3 x^2) (-1 + y) T[x, y]^2 - 
  x (-1 + x^2) (-2 + y) T[x, y] D[T[x, y], x] + 
  x (-1 + x^2) (-1 + y) y D[T[x, y], y] D[T[x, y], x], 
 x (-1 + x^2) T[x, y] D[S[x, y], x] + 
  S[x, y] ((2 - 6 x^2) T[x, y] + x (-1 + x^2) D[T[x, y], x])};

where T[x,y], S[x,y] are some unknown funcitons. We want to solve the equations around y=0 in a series expansion, so we define

T[x_, y_] = Sum[tx[jj, x]*y^jj, {jj, 0, 7}];
S[x_, y_] = Sum[sx[jj, x]*y^jj, {jj, 0, 7}];

and then to first order we have

Simplify[Series[expr, {y, 0, 0}]]

We see that we can solve the first equation with no problem

DSolve[
 SeriesCoefficient[Simplify[Series[expr, {y, 0, 0}]], 0][[1]] == 
  0, {tx[0, x]}, x]

{{tx[0, x] -> x^2 (1 - x^2)^2 C[1]}}

But then when we plug that solution back in, it doesn't quite work

Simplify[Series[
   expr, {y, 0, 0}] /. {tx -> Function[x, x^2 (1 - x^2)^2 C1]}]

What am I doing wrong? My question is basically how to enter in Mathematica objects like $a_n(x)$ (and also $a_n(x,y)$ or even $a_{n,m}(x,y)$) where $n$, ($m$) is a non-negative Integer (a dummy index) and $x$, ($y$) is a variable (Real/Complex depending on the problem), so that we can manipulate them like any other function.

Answer 1

First you can try Subscript (Control key + _) to make tx and ty look more like a coefficient, then use Subscript[tx,0] -> Function[{x,y},x^2 (1-x^2) C1]. The reason for making the function of x and y even when it is a function only of x is to convince the partial derivative function that its y derivative is 0. Another way to do similar things is to use SetAttributes and Constant, but I don’t know whether it can make eg ‘constant only respect to y’.

Answer 2

Probably not complete answer, but some (clumsy) thoughts.

The idea is to introduce a dummy variable instead of y (=0) so that Mathematica can properly compute partial derivatives:

expr0 = Normal@
   Series[expr, {y, 0, 0}] /. {(t : (T | S))[x, 0] :> t[x, z], 
   Derivative[n_, m_][t : (T | S)][x, 0] -> Derivative[n, m][t][x, z]}

{-4*(-1 + 3*x^2)*T[x, z]^2 + 2*x*(-1 + x^2)*T[x, z]*Derivative[1, 0][T][x, z], x*(-1 + x^2)*T[x, z]*Derivative[1, 0][S][x, z] + S[x, z]*((2 - 6*x^2)*T[x, z] + x*(-1 + x^2)*Derivative[1, 0][T][x, z])}

All the next is simple enough:

tsol = DSolve[expr0[[1]] == 0, T[x, z], x]

{{T[x, z] -> 0}, {T[x, z] -> x^2 (1 - x^2)^2 C[1]}}

And for the second variable:

Block[{T}, T[x_, z_] = tsol[[2, 1, 2]]; 
 DSolve[expr0[[2]] == 0, S[x, z], x]]

{{S[x, z] -> C[2]}}