Differential equation of compounded interest I came across this explanation and don't understand how the equation came to be.
Let I represent the amount of money deposited in bank, r the interest rate (compounded) and t represent time.
Then the value of risk-free investment increases as $I(t+dt) = I(t)+dI$
So far so good.
And then it says that that $dI = rIdt$ provided $dt$ is any small finite increment.
Why is it rIdt? Why multiply I by r? Shouldn't it rather be $((r+1)^{dt}-1)I$ since we're talking about compounded interest?
The textbook then goes on about solving the separable first order differential equation and ending up with the solution $I(t) = I_0 e^{r(t-t_0)}$

Note that $dI$ represent the change in your account during $dt$ time. This is rate times the balance times the time. $$dI=rI dt$$ 
It is a linear approximation but since $dt$ is infinitesimal we use linear approximation and solve the differential equation to get the answer. 
The solution is of course exponential as expected.
