Modeling Differential equations?

Question

Ok, this is a somewhat complicated first order equation but if you can help me out, it'll be a big help. A recent college graduate borrowed $100,000 at an interest rate of 9% to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of 800(1 + t/120), where t is the number of months since the loan was made. a.) Assuming that this payment schedule is maintained, when will the loan be fully paid? ANSWER = 135.36 months. b.) Assuming the same painment schedule how large can the loan be paid off in exactly 20 years? ANSWER = $152698.56. How do you get this?

Answer 1

Look at level-interest loans in general. Loans are compound interest investments made by the bank. You pay back (for the 9 % loan), 3/4% of the unpaid balance each month plus a certain amount of principal so that all payments will be uniform (or nearly so) and that there are a stated number of monthly payments. There is a formula for this, and the results are tabulated.

In your version, the person starts paying $ 800 a month. In the first month, the interest is $ 750, the principle paid back $50, and the unpaid balance drops to $ 99,950. The second month, the interest is 0.00075*$99,950, and the person pays 1.00833*800. This loan get paid in 135+ months. In the second problem, you borrow more, so that the loan takes 240 months to pay off.

A differential equation is NOT required to do these problems. In fact, since they are not continuous functions, a differential equation is not appropriate. The heuristic can be easily set-up on a speadsheet, although the solution may require some trial-and-error.