Continuous Monthly Payments

I am just beginning differential equations, so this is probably a very simple question, please forgive my ignorance! Here is the question:

$12,000 is borrowed. The loan is to be paid in 60 equal monthly installments at an interest rate of 5% per year. If the payments are actually made continuously at the necessary rate to pay off the loan in 60 months. Determine the continuous rate per month that would be required.

I believe the debt should grow using this equation:

y = value of the debt
t = time
(1/y)(dy/dt) = k

However, I do not know how to simultaneously consider the continuous monthly payments forcing the debt downward.

Also, if anyone knows a good resource on modeling situations with differential equations, please let me know because my textbook does not describe this very well.

Any help would be appreciated.

Quote:

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Originally Posted by machi4velli

I am just beginning differential equations, so this is probably a very simple question, please forgive my ignorance! Here is the question:

$12,000 is borrowed. The loan is to be paid in 60 equal monthly installments at an interest rate of 5% per year. If the payments are actually made continuously at the necessary rate to pay off the loan in 60 months. Determine the continuous rate per month that would be required.

I believe the debt should grow using this equation:

y = value of the debt
t = time
(1/y)(dy/dt) = k

However, I do not know how to simultaneously consider the continuous monthly payments forcing the debt downward.

Also, if anyone knows a good resource on modeling situations with differential equations, please let me know because my textbook does not describe this very well.

Any help would be appreciated.

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The rate of change of debt y has two components the interest =ky and
minus the repayment per unit time =-c, so the rate of change of debt is:

dy/dt=ky-c

where k=ln(1+rt/100) when t is measured in years, and rt is the annual
interest rate in percent. Now you need to solve the DE with boundary
conditions that y(0)=L the initial loan, and y(5)=0 (equivalent to the loan
being paid off after 5 years = 60 months).

RonL

Well, that was pretty simple, but how did you find k? Don't I need to isolate dy/y so that I can integrate?

Quote:

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Originally Posted by machi4velli

Well, that was pretty simple, but how did you find k? Don't I need to isolate dy/y so that I can integrate?

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I found k by considering the situation with no repayments, then after
1 year the debt is (1+rt/100) times the original loan.

So we solve dy/dt=ky to get y(t) (assume that the initial loan is 1),
then y(1)=(1+rt/100).

RonL

After 1 year, the debt should be

1.05 * (12000 - 12*(the monthly payment))

??

Edit: nvm, now I know what to do, thanks