# Log transforming an ODE

I'm doing some numerical simulations of an exponential growth like system which, for simplicity, has the form:

$$\frac{dx}{dt}= ax + bxy \quad\quad \frac{dy}{dt}= cy + dxy$$

For some parameter values i get instability in the simulation though I remember reading a paper which used log transformations to prevent this. Any ideas on how I could do this or how to rewrite the equations as:

$$\frac{d log(x)}{dt}= \ldots \quad\quad \frac{d log(y)}{dt}= \ldots$$

## 1 Answer

(If you need more information, for example Lyapunov functions, this equation is similar to Lotka-Volterra equation.)

Dividing the first equality by $x$: $$\frac{1}{x} \frac{dx}{dt}=a+by$$ i.e: $$\frac{d \log(x)}{dt}=a+by$$ and similarly: $$\frac{d \log(y)}{dt}=c+bx$$

• Thanks for the answer. How do you go from $\frac{1}{x} \frac{dx}{dt}$ to $\frac{d log(x)}{dt}$? – Tom Sep 10 '18 at 19:28
• @Tom It can also be written as: $\frac{d}{dt} \log(x(t)) = \frac{1}{x(t)}x'(t)$ – Coolwater Sep 11 '18 at 10:44