How to solve this system of nonlinear ODEs

Question

I have a system of two nonlinear ODEs:

$${dk\over dt}= -\epsilon \tag{1}$$ and $${d\epsilon\over dt} = -c{\epsilon^2\over k} \tag{2}$$

Where $c$ is a constant.

I tried to differentiate $(1)$ w.r.t to time and then substitute it in (2):

$$-{d^2 k\over dt^2} = -c {\left({dk\over dt}\right)^2\over k}$$ Or

$$k{d^2k\over dt^2} = c\left({dk\over dt}\right)^2 \tag{3}$$

But now, I have no idea how to proceed to solve this. Could you please point me how to solve this system of ODEs?

I appreciate your help

Answer 1

Hint.

$$ \frac{k''}{k'}=c\frac{k'}{k} $$

or

$$ \ln(k')'=c\ln(k)' $$

or

$$ k' = C_0k^c $$