Modelling interest with differential equations (Interpretation)

Question

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows:

Assume you have a bank account that grows at an annual interest rate of r and every year you withdraw a fixed amount from the account (denoted w). Assuming continuous compounding interest and continuous withdrawal, the described account follows the differential equation:

${\frac{dP}{dt} = rP - w}$

Where P(t) is the amount of money in the account at time t.

I am confused as to why we subtract the entire amount ${w}$ in the equation? My interpretation (which is obviously incorrect) is that ${\frac{dP}{dt}}$ is the change in the amount of money in the account with respect to time for any given time ${t}$. If this is so, wouldn't the above equation mean that at every instant ${t}$ we are adding ${rP}$ to the account and subtracting the entire annual deduction ${w}$ so that at the end of an entire year we've deducted more than ${w}$ from the account?

Answer 1

$r$ is a rate of interest, meaning a percentage per unit time. So say $r$ is 5% per year, the instantatneous rate of growth of the account is 5% of the amount in the account per year: $rP$.

Similarly you have to interpret $w$ as the withdrawal rate per unit time, for example, $1,000 per year.

You are right that the equations are meaningless if yo take $w$ to mean that every instant you withdraw the finite amount $w$. But that is not what is meant.

Answer 2

$w$ is the (continuous, constant) rate of withdrawal, and $rP$ is the (continuous, proportional) rate of interest accrual.

Answer 3

$\frac{dP}{dt}$ measures the rate of increase of $P$ in currency units per year

At any instant the total in the account is increasing at a rate of $rP$ per year and decreasing at a rate of $w$ per year.