# Help with first order non-linear differential equation

I've been trying to solve this one for a while, but I still can't make it. Here's the problem.

I have a $$f_1(t;\rho,\nu)$$ that for $$t\to\infty$$ and for $$\rho>\nu$$ goes as $$f_1\sim t^{\nu/\rho}.$$

Saying that, the problem is with a second function $$f_2(t, \rho\, \nu)$$ which obeys the equation:

$$\frac{df_2(t)}{dt}=\frac{\nu f_2(t) + (\nu+1)f_1(t)}{\rho t + (\nu+1)[f_1(t)+f_2(t)]}$$

with $$\rho,\nu\in\mathbb{N}^+$$ and $$f_2(t)>0$$ (and increasing).

I am interested in the long term behavior as for $$f_1$$. I know (from simulations) that for $$t\to\infty$$ it behaves similarly to $$f_1$$, but with a different exponent. I've tried to plug in the $$f_1$$ written above (assuming $$\rho>\nu)$$ and also to force $$f_2(t)\sim ct^\alpha$$ as a solution, ultimately looking for an expression for $$\alpha$$ as a function of the other parameters, but I only arrived to contradict myself.

I've also tried on Mathematica with initial conditions $$f_2(0)=0$$ but without succeeding.

Any tips?

• If this is a differential equation, you don't want $t \in \mathbb N$. – Robert Israel Mar 29 at 15:54
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $$f_2(t)$$, I tried plugging in a form for $$f_2(t)$$ and solving for $$f_1(t)$$. I found that $$f_2(t) = \frac{\nu + 1}{\rho} \log(t)\; t^{\nu/\rho}$$ is a solution with $$f_1(t) = {\frac {1}{\rho} \left( {t}^{{\frac {\nu+ \rho}{\rho}}}{\rho}^{3}+\ln \left( t \right) {t}^{2\,{\frac {\nu}{ \rho}}} \left( \nu+1 \right) ^{2} \left( \ln \left( t \right) \nu+ \rho \right) \right) \left( - \left( \nu+1 \right) \left( \ln \left( t \right) \nu+\rho \right) {t}^{{\frac {\nu}{\rho}}}+{\rho}^{2 }t \right) ^{-1}} \sim t^{\nu/\rho}$$
• Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^\alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$ – Meffo Apr 2 at 10:55
• No, I plugged in $f_2(t) = c \log(t) t^{\nu/\rho}$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) \sim t^{\nu/\rho}$. – Robert Israel Apr 2 at 12:00