Let $u_1(t), \cdots, u_n(t)$ defined on $\mathbb R$ and let $W(t:u_1, u_2, \cdots, u_n)$ stand for Wronskian $\begin{vmatrix} u_1(t) & \cdots & u_n(t) \\ u_1'(t) & \cdots & u_n'(t) \\ \vdots & & \vdots \\ u_1^{(n-1)} (t)& \cdots & u_n^{(n-1)} (t) \end{vmatrix}$.
Suppose $W(t: u_1, \cdots, u_n)\neq 0.$
If $n-1$ times differentiable function $v(t)$ satisfies $v^{(k)}(t)=c_1u_1^{(k)}(t)+\cdots + c_nu_n^{(k)}(t)$ for $k=0,1,\cdots, n-1$, then prove that $c_i=\dfrac{W(t:u_1, \cdots, u_{i-1}, v, u_{i+1}, \cdots, u_n)}{W(t:u_1, \cdots, u_n)} \ (i=1, \cdots, n)$
I could calculate this in the easier case $n=2$, but I don't know what I have to do in the case of $n$.
How can I calculate $c_i$ ?