I have two positive functions $f(t,a),g(t,a)$ related by an ODE system of form $$\dfrac{df}{dt}=F_1(f(t,a),g(t,a),a),\\ \dfrac{dg}{dt}=F_2(f(t,a),g(t,a),a),$$
where $a$ is a parameter.
I know that for all fixed $a$ I have the same curve $\mathcal{C}$ for the parametrization $(f(t,a),g(t,a))$ for $t\in(0,\infty)$, I mean, I have the same phase-plane.
I'd like to prove that the velocity in which the curve $\mathcal{C}$ is traveled is greater for $a$ bigger.
However, the calculations are not possible to be explicited.
I know that, given positive fixed functions $p,q$, I have $F(p(t,a),q(t,a),a)>F(p(t,b),q(t,b),b)$ for all time $t$, in each coordinate, if $a>b$.
Thank you so much for any suggestion to procedure.
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A very simple example.
$$\dfrac{df}{dt}=a/f(t),\\ \dfrac{dg}{dt}=a,$$
We see that $dg/df$ is invariant for $a$, so the curve $(f,g)$ is invariant for $a$.
The velocity squared is $a^2\dfrac{f^2+1}{f^2}$. How can I calculate the derivative wrt $a$ to see how the velocities of the parametrizations change if $f$ is also related with $a$? (In this case, the system can be solved explicitly, but I am seeking a general procedure!)