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