Formal recursion for solved analytic normal equations

Write $u={\sum }_{\mathrm{\ell }\ge 0}{a}_{\mathrm{\ell }}(x)t^{\mathrm{\ell }}$ with $a_{\mathrm{\ell }}\in \mathbb{R}[[x]]$. The initial conditions force $a_0=\cdots =a_{m-1}=0$. Consequently each allowed jet has a factor $t^{m-j}$ and in particular zero total constant term. F1 makes substitution into F a well-defined formal series in x,t.

For $q\ge 0$, equating coefficients gives $\frac{(q+m)!}{q!}{a}_{q+m}(x)=[t^q]F(t,x,({\mathrm{\partial }}_x^{\alpha }{\mathrm{\partial }}_t^{j}u))$. In the right side, a coefficient of t-degree at most q in an allowed jet uses only $a_{\mathrm{\ell }}$ with $\mathrm{\ell }-j\le q$, hence $\mathrm{\ell }\le q+m-1$. Tangential differentiation changes x-degrees but never t-degree. Replacing u by its truncation through that index therefore leaves the coefficient unchanged.

Starting at q=0, step 2.1 defines each new $a_{q+m}$ by division by the nonzero integer $(q+m)!\mathrm{/}q!$. Every right-side coefficient is thus eventually matched, proving existence of a formal solution. The same recursion forces every coefficient of any other formal solution, proving uniqueness. For m=1 and a finite vector, apply the same recursion simultaneously to all components; no next coefficient of any component occurs on the right. F2 supplies the analytic-data reduction.