方程一

已知\(C, D​\)都是长度为\(n​\)的多项式,求\(F​\), \(F′=Ce^F+D \pmod {x^n}​\)

Sol:

\[ \begin{aligned} F' = G(F) &= Ce^F + D \\ &= G(F_0) + G'(F_0) (F - F_0) \\ &= Ce^{F_0} + D + Ce^{F_0}(F - F_0) \\ &= TF + Z \end{aligned} \]

\[ \begin{aligned} 设U' = TU, \frac{dU}{dx} &= TU \\ \ln(U) &= \int T dx \\ U &= e^{\int Tdx} \end{aligned} \]

\[ \begin{aligned} 设F = UV, (UV)' &= TUV + Z \\ UV' + VU' &= U'V + Z\\ V &= \int \frac {Z}{U} \end{aligned} \]

方程二

\[ \begin {aligned} F &= \int e^{T-F}dx \\ e^F F' &= e^T \\ e^F &= \int e^T + 1 \\ F &= \ln\left (\int e^T + 1\right) \end {aligned} \]

留坑链式反应