Exercise 5, § 3

Functions of a Real Variable, I, 40

Let $u$ and $v$ be two real functions which are $n$ times differentiable on an interval $I \subset \mathbf{R}$ . If one puts $D^n(u/v) = (-1)^n w_n / v^{n+1}$ at every point where $v(x) \neq 0$ , show that

w_n = \begin{vmatrix} u & v & 0 & 0 & \ldots & 0 \\ u' & v' & v & 0 & \ldots & 0 \\ u'' & v'' & 2v' & v & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ u^{(n-1)} & v^{(n-1)} & \binom{n-1}{1} v^{(n-2)} & \binom{n-1}{2} v^{(n-3)} & \ldots & v \\ u^{(n)} & v^{(n)} & \binom{n}{1} v^{(n-1)} & \binom{n}{2} v^{(n-2)} & \ldots & \binom{n}{n-1} v' \end{vmatrix}

(put $w = u/v$ and differentiate $n$ times the relation $u = wv$ ).

Solution

No solution has been written yet.