Differential equation for bank balance

[Xamfy19]

Hello, I need help for the following question:
Peter borrowed $100,000 from bank and he pays back at a rate of $12,000 per year. The bank charges him interest at a rate of 7.25% per year compounded continuously. Make a continuous model of his situation using differential equation involving dB/dt where B = B(t) is the balance he owes the bank at time t.

I thought the B(t) is something like
B(t) = (100000-12000t) + Sum(n from 1 to t) [100000-12000(n-1)]*0.0725.

But, I am not sure yet. Thanks for help...

 

[courtrigrad]

$$B(t) = B_{0}e^{rt}$$
$$\frac{dB}{dt} = rB_{0}e^{rt} = rB(t)$$

 

[Xamfy19]

Thanks, but how the payback rate is taken into account?

 

[HallsofIvy]

The point of writing it as a differential equation- i.e. a differentiable function, is to avoid having to take into account individual payments as you do with your sum. Treat the problem as if money is being paid back continuously through the year (at a rate of 12000 per year) while interest is accumulating continously (and so at (e<sup>0.0725t[/itex]- 1)B).

$$\frac{dB}{dt}= (e^{0.075t}- 1)B- 12000$$
with initial value B(0)= 100000.</sup>