Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$, $f$ is analytic on $\mathbb{R^n}$. $(x_1(t),x_2(t),...,x_n(t))$ is an unknown curve passing through $(0,0,...,0)$.$t\in\mathbb{R}$ is "time".
A constraint for $f$ is:
The formula $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ should be constantly $(0,0,...,0)$ along the curve $(x_1(t),x_2(t),...,x_n(t))$.
I want to know about the curve $(x_1(t),x_2(t),...,x_n(t))$. For example,when do they exist,when are they unique,or can I derive the exact curve, and many other aspects.
Here are what I am intend to do to solve the problem:
Since $f$ is constantly $(0,0,...,0)$,I calculate the $n$th derivation of $f$:
The first derivation is:
$A\cdot X'$ where $A$ is the Jacobian of $f$ and $X'$ is the column vector $(\frac{dx_1}{dt},...,\frac{dx_n}{dt})^T$.Thus,and $X'$ can be solved by the linear equation $A\cdot X'=0$.
I can get the other $n$th derivation by the law of derivation of Matrix: (Given Matrix $S$ and $T$,$(S\cdot T)'=S'\cdot T=S\cdot T'$.)
So then the 2nd derivation is: $A'\cdot X+A\cdot X'$,notice that $A'$ is not an exact constant matrix but contains $X'$ term.While $X'$ can derived by the linear equation $A\cdot X'=0$.
Keep doing this,a block-matrix can be derived:
$\begin{bmatrix} b_{11} A&0 &0 & 0 &...&0 \\ b_{21}A'&b_{22} A &0 &0&... &0 \\ b_{31}A''&b_{32} A' &b_{33}A &0&... &0 \\ b_{41}A^{(3)}&b_{42}A'' &b_{43}A' &b_{44}A&... &0 \\ ...\\ b_{n1}A^{(n)}&b_{n2}A^{(n-1)}&&&...&0 \end{bmatrix}\cdot \begin{bmatrix} X' \\ X''\\ X^{(3)}\\ ...\\ X^{(n)} \end{bmatrix}=\begin{bmatrix} 0 \\ 0\\ 0\\ ...\\ 0 \end{bmatrix}$
$b_{i,j}$ are constants. Usually,Given any specific $f$,Each $X^{(i)}$ can be derived by the $i$th equation. But is there any thing can I derive when $f$ is not specific but with parameters?Such as the existence or the uniqueness of the curve $(x_1(t),x_2(t),...,x_n(t))$？
Thanks ahead!
