Exercise 9, § 2

Let $y_k$ ($1 \leq k \leq n$) be $n$ real functions defined on an interval $I \subset \mathbf{R}$ and having a continuous $(n-1)^{th}$ derivative.

a) Show that if the $n$ functions $y_k$ are linearly dependent then the matrix $(y_k^{(h)}(t))_{0 \leq h \leq n-1 \atop 1 \leq k \leq n}$ has rank $< n$ at every point $t \in I$.

b) Conversely, if for every $t \in I$ the matrix $(y_k^{(h)}(t))_{0 \leq h \leq n-1 \atop 1 \leq k \leq n}$ has rank $< n$, show that in every nonempty open interval $J \subset I$ there exists a nonempty open interval $U \subset J$ such that the restrictions of the $y_k$ to $U$ are linearly dependent (if $p$ is the smallest of the numbers $q < n$ such that the Wronskians of any $q$ of the functions $y_k$ are identically zero on $J$, consider a point $a \in J$ where the Wronskian of $p-1$ of the functions $y_k$ is not zero, and show that $p$ of the functions $y_k$ are integrals of a linear equation of order $p-1$ on a neighbourhood of $a$).

c) Put $y_1(t) = t^2$ for $t \in \mathbf{R}$, $y_2(t) = t^2$ for $t \geq 0$, $y_2(t) = -t^2$ for $t < 0$; show that $y_1$ and $y_2$ have continuous derivatives on $\mathbf{R}$ and that $y_1 y_2' - y_2 y_1' = 0$, but that $y_1$ and $y_2$ are not linearly dependent on $\mathbf{R}$.