Differential equation for bank balance

In summary, Peter has borrowed $100,000 from a bank and is paying back at a rate of $12,000 per year. The bank charges him interest at a rate of 7.25% per year compounded continuously. A continuous model of his situation can be represented by the differential equation dB/dt = (e^0.0725t - 1)B - 12000. This takes into account the continuous payback rate and accumulating interest, with an initial balance of $100,000.
  • #1
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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...
 
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  • #2
[tex] B(t) = B_{0}e^{rt} [/tex]
[tex] \frac{dB}{dt} = rB_{0}e^{rt} = rB(t) [/tex]
 
  • #3
Thanks, but how the payback rate is taken into account?
 
  • #4
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 (e0.0725t[/itex]- 1)B).

[tex]\frac{dB}{dt}= (e^{0.075t}- 1)B- 12000[/tex]
with initial value B(0)= 100000.
 

1. What is a differential equation for bank balance?

A differential equation for bank balance is a mathematical model that describes the change in a person's bank balance over time. It takes into account factors such as income, expenses, interest rates, and other financial transactions.

2. Why is a differential equation needed for bank balance?

A differential equation is needed for bank balance because it allows us to predict how a person's bank balance will change over time, and to make informed decisions about spending and saving. It also helps us understand the underlying factors that contribute to changes in bank balance.

3. How is a differential equation for bank balance solved?

A differential equation for bank balance is typically solved using mathematical techniques such as separation of variables, substitution, or using computer software. The solution provides an equation that describes the relationship between bank balance and time.

4. Can a differential equation for bank balance accurately predict future balances?

While a differential equation can provide a general understanding of how a person's bank balance may change over time, it is not a perfect predictor. Many external factors, such as unexpected expenses or changes in income, can affect a person's actual bank balance.

5. Are there any limitations to using a differential equation for bank balance?

Yes, there are limitations to using a differential equation for bank balance. It assumes that all variables, such as income and expenses, remain constant over time. It also does not account for unpredictable events or changes in financial habits. Additionally, the accuracy of the prediction may decrease over longer periods of time.

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