# If I rescale the time in a differential equation, do I need to adjust the parameters?

Imagine I have a differential equation and I have some data and the model is supposed to fit the data. If I now rescale the time in the range 0 to 1, do I need to adjust the parameters of the equations? How would one adjust them?

• I’m voting to close this question because it's off topic for computational science but might be appropriate in math.stackexchange. – Brian Borchers Nov 18 '20 at 0:56
• Actually, if we could see the structure of the equation it would be clearer if this question belongs more to math or to computational science – Maxim Umansky Nov 18 '20 at 3:03

Yes. Let's assume you have an ODE of the form $$x'(t) = kx(t)$$ and that your coefficient is $$k=42$$. If the physical units of $$x$$ are meters and of $$t$$ seconds, then what that really means is that $$k=42 \frac{1}{s}$$.
So now if you rescale time -- say, you want to measure time in minutes, you still have $$k=42 \frac{1}{s}$$ but you want to express this also in minutes, which would make it $$k=42 \frac{1}{s} = 42 \frac{1}{\frac{1}{60}\text{min}} = 42\times 60 \frac{1}{\text{min}} = 2520 \frac{1}{\text{min}}.$$ That means, the coefficient is exactly the same as before, but it has a different numeric value that would have to be input into a program.