Let $f(x,y),g(x,y)$ be two real-analytic functions near a neighbourhood of $(0,0)$. Consider the following Wronskian determinant: $$ D(x,y):=\det \begin{bmatrix} f(x,y) & g(x,y)\\ \partial_x f(x,y) & \partial_x g(x,y) \end{bmatrix}. $$ If that $D(x,y)=0$ constantly, we know that for each $y$, $f$ and $g$ are linearly dependent when viewed as functions of $x$. That means there are functions $\lambda(y)$ and $\mu(y)$, not simultaneously zero at any $y$, such that for every $(x,y)$ we have $$ \lambda(y)f(x,y)+\mu(y)g(x,y)= 0. $$ My question is:
- Can we always normalise so that we may assume $\lambda(y)=\cos \theta(y)$ and $\mu(y)=\sin \theta(y)$?
- Are $\lambda(y)$ and $\mu(y)$ smooth, or even real-analytic?