Does there exist two functions $f, g\in C^1(I)$ for which $W(f, g) (x) >0$ for some $x$ and $W(f, g) (x) <0$ for some $x$?

Question

$f, g\in C^1(I) $ where $I$ is an open interval and $f, g$ both are real valued.

Let $W(f,g)(x) =\begin{vmatrix}f(x) &g(x) \\f'(x)&g'(x)\end{vmatrix}$ denote the Wronskian of $f, g$ at $x\in I$

$\textbf{Question}$ Does there exists such $f, g$ for which $W(f, g) (x) >0$ for some $x$ and $W(f, g) (x) <0$ for some $x$ ?

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$W(f, g) (x) \neq 0$ for some $x\in I$ implies $\{f, g\}$ linearly independent.

If two functions are solutions of a differential equation $y"+p(x) y'+q(x) y=0$ on $I$ where $p, q\in C(I) $ then by Abel's identity we have

$$W(f, g) (x) =W(f, g) (x_o) e^{-\int_{x_0}^{x} p(t) dt}$$

Then $W(f, g) (x_0) \neq 0$ for some $x_0\in I$ implies $W(f, g) \neq 0$ on $I$

Moreover $W(f,g)$ different from zero with the same sign at every point ${\displaystyle x} \in {\displaystyle I}$

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Hence we have to find two functions $f, g$ with the properties:

1. $f, g$ must have to be linearly independent.

2. $f, g\in C^1(I) $

3. $f, g$ can't be the solution of $2$nd order homogenous linear ODE.

4. $W(f, g) $ attains both positive and negative values on $I$.

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Let $0\in I$ be an open interval and $f, g\in C^1(I) $ defined by $f(x) =x^2$ and $g(x) =x|x| $

Then $f,g$ satisfy $1, 2,3 $ but not $4$ as $W(f, g) (x) =0$ on $I$.

Answer 1

Yes, that is possible: For $f, g\in C^1(I)$ with $g(x) \ne 0$ on $I$ has $$ W(g, fg)(x) = g(x)^2 \cdot W(1, f)(x) = g(x)^2 f'(x) $$ the same sign as $f'(x)$, that allows the easy construction of examples such that the Wronskian takes both positive and negative values.

Some concrete examples are

• $W(1, x^2) = 2x$,
• $W(e^x, e^x \sin(x)) = e^{2x} \cos(x)$.