I have linearised some equations and trying to solve them perturbatively in powers of small parameter $e$. Here is my script
Subscript[t, x] = 1 - (Subscript[B, x]^2 e^2)/2 + (7 Subscript[B, x]^4 e^4)/8; Subscript[t, y] = Subscript[B, x] e - Subscript[B, y]^3 e^3; Subscript[n, x] = -Subscript[t, y]; Subscript[n, y] = Subscript[t, x]; Subscript[\[Sigma], xx] = -d + 2 \[Mu] Subscript[u, x]; Subscript[\[Sigma], yy] = -d + 2 \[Mu] Subscript[v, y] e^-2; Subscript[\[Sigma], xy] = \[Mu] ( Subscript[v, x] e^-1 + Subscript[u, y] e^-1); Subscript[u, x] = Subscript[u, x0] + Subscript[u, x2] e^2 + Subscript[u, x4] e^4; Subscript[u, y] = Subscript[u, y0] + Subscript[u, y2] e^2 + Subscript[u, y4] e^4; Subscript[v, x] = Subscript[v, x0] + Subscript[v, x2] e^2 + Subscript[v, x4] e^4; Subscript[v, y] = Subscript[v, y0] + Subscript[v, y2] e^2 + Subscript[v, y4] e^4; d = d0 + d2 e^2 + d4 e^4; Series[FullSimplify[ e (Subscript[\[Sigma], xx] Subscript[n, x] + Subscript[\[Sigma], xy] Subscript[n, y])]==0, {e, 0, 4}] Series[FullSimplify[ e^2 (Subscript[\[Sigma], yy] Subscript[n, y] + Subscript[\[Sigma], xy] Subscript[n, x])]==0, {e, 0, 4}] This outputs two equations that I need to solve. In the first order $e^0$, these equations simplify to
(Subscript[u, y0] + Subscript[v, x0])=0 and
`2 \[Mu] Subscript[v, y0]=0` Which mean I can use $2 \mu v_{y0} =0$ and $(u_{y0}=- v_{x0})$ to simplify the higher order equations $e^2$ and $e^4$. But how can I do this without manually subbing lower order equations to higher order equations?
e1ande2, one can doSolveAlways[{e1, e2}, e]. $\endgroup$