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I hope someone can help me with the following question.

Find an explicit solution to the following problem

$$\dot{u}(t)=\frac{k_1 u(t)}{u_* + u(t)}-k_2 u(t)$$

with initial condition $u(0)=u_0$.

Now I tried solving this via separation of variables, but I get following equation:

$$k_1 \ln \left|\frac{k_2 u(t)+k_2 u_* -k_1}{k_2 u_0 + k_2 u_* -k_1}\right|-k_2 u_* \ln \left|\frac{u(t)}{u_0}\right|=-tk_2(k_2 u_*-k_1)$$

I cannot solve this equation explicity, because there is always a nonlinear term with the $u(t)$.

Can someone give me a hint on what I am doing wrong?


Thanks in advance and best regards,

silcrystal

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  • $\begingroup$ What is $u_*$? Is it a constant? $\endgroup$ – Peter Foreman Apr 7 at 19:49
  • $\begingroup$ Yes, $k_1,k_2,u_*>0$ are constants. $\endgroup$ – silcrystal Apr 7 at 19:51
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The equation can be written in the form

$$\left(a+\frac b{cu+d}\right)\frac{\dot u}u=1$$

which integrates as

$$e\log|u|+f\ln|cu+d|=t+g.$$

The LHS can indeed not be inverted.

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  • $\begingroup$ Thank you. So how can I analyse the behaviour of the solution if don't have it explicitly given? $\endgroup$ – silcrystal Apr 8 at 7:42
  • $\begingroup$ @silcrystal: analyse $t(u)$. $\endgroup$ – Yves Daoust Apr 8 at 8:44
  • $\begingroup$ Ah that's a good idea, thank you very much. $\endgroup$ – silcrystal Apr 8 at 9:55

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