Explicit solution to a nonlinear ODE

Question

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

Answer 1

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.