If you have a simple situation involving two variables $x$ and $t$. As $t$ increases $x$ decreases but is positive and you don't know the direct relationship between them so you create a differential equation. Obviously you can't say $$\frac{dx}{dt}=x$$ which says the value of the function is equal to the slope because in our situation the slope is negative.

So then you assume $$\frac{dx}{dt}=-kx$$ but how do you know this is going to give the function with the characteristics above ?

Could it not give a function with negative values and positive slope ?


  • $\begingroup$ It could and sometimes it does: provided $k>0$, any solution to $x'(t)=-kx(t)$ with $x(0)<0$ will be negative for all $t$, with strictly positive derivative. $\endgroup$ – Saucy O'Path May 12 at 22:52

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