Find two LI continuous differentiable functions $f,g: I \in \mathbb{R} \rightarrow \mathbb{R}$ such that $(f\cdot g')(t) - (f'\cdot g)(t)) = 0$.

Question

Find two linear independent continuous differentiable functions $f,g: I \in \mathbb{R} \rightarrow \mathbb{R}$ with $(f\cdot g')(t) - (f'\cdot g)(t)) = 0$.

Any idea on how can I find such functions? I imagine that, if $(f\cdot g')(t) - (f'\cdot g)(t)) = 0$, then:

$(f\cdot g')(t) = (f'\cdot g)(t)) \Longrightarrow \frac{f'}{f}(t) = \frac{g'}{g}(t) \Longrightarrow \ln(f(t)) = \ln(g(t)) + C \Longrightarrow f(t) = C \cdot g(t)$.

This implies $f,g$ are L.D., a contradiction. Maybe the trick is with choosing the right interval $I$ to define the functions. I don't know.

Any help? Thanks!

Answer 1

Hint: Let $I=[-1,1]$. $f(t)=t^2$, $g(t)=t\left|t\right|$.