含exp关于(u,v)的参数(x,y)的方程组 渐近解

hbghlyj

当$(u,v)≈(0,0),(x,y)≈(0,0)$, 求1阶渐近解
\begin{cases}\exp (3 u)+\exp (4 v)+\exp (x)+\exp (2 y)=4\\\exp (u)+\exp (v)+\exp (x)+\exp (y)=4\end{cases}Solution.
把$\exp(x)$替换为$1+x$
\begin{cases}1 + x + 1 + 2 y + 1 + 3 u + 1 + 4 v = 4 \\
1 + x + 1 + y + 1 + u + 1 + v = 4\end{cases}解得$\cases{u = -3 x - 2 y\\v =2 x + y}$

用AsymptoticSolve验证:

1. AsymptoticSolve[
2. Exp[x]+Exp[2y]+Exp[3u]+Exp[4v]==4 &&
3. Exp[x]+Exp[y]+Exp[u]+Exp[v]==4,
4. {{u, v}, {0, 0}}, {{x, y}, {0, 0},1}]

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输出{{u -> -3 x - 2 y, v -> 2 x + y}}

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等价于$\begin{cases}\exp (3 u)+\exp (4 v)=4-\exp (x)-\exp (2 y)\\\exp (u)+\exp (v)=4-\exp (x)-\exp (y)\end{cases}$
要解出$u,v$

用Gradient一阶近似$f(x)=y_{0}+D f\left(x_{0}\right)\left(x-x_{0}\right)+o\left(\left|x-x_{0}\right|\right)$, 假设余项为0, 则有
$$y=f(x) \quad \Longleftrightarrow \quad x=x_{0}+D f\left(x_{0}\right)^{-1}\left(y-y_{0}\right)$$

验证

1. f={Exp[3u]+Exp[4v]+Exp[x]+Exp[2y],Exp[u]+Exp[v]+Exp[x]+Exp[y]};
2. G=D[f,{{u,v}}]/.{u->0,v->0};
3. Y={{2-Exp[x]-Exp[2y]},{2-Exp[x]-Exp[y]}};
4. Normal@Series[Normal@Series[Inverse[G].Y,{x,0,1}],{y,0,1}]

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hbghlyj

用AsymptoticSolve得到的答案是$\begin{cases}u=-3 x-2 y&+45 x^2+54 x y+16 y^2\\v=2 x+y&-52 x^2-62 x y-19 y^2\end{cases}$

1. AsymptoticSolve[
2. Exp[x]+Exp[2y]+Exp[3u]+Exp[4v]==4 &&
3. Exp[x]+Exp[y]+Exp[u]+Exp[v]==4,
4. {{u, v}, {0, 0}}, {{x, y}, {0, 0},2}]

复制代码

hbghlyj

$+45,+54,+16$
$-52,-62,-19$
这些是怎么筭出的