Tutorial3

Tutorial3 - Applications: Modeling with First Order...

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Applications: Modeling with First Order Equations In this note we present examples of a few simple mathematical models with first order differential equations, like compound interest with deposits or withdrawals, velocity of falling mass, escape velocity and so on. Construc- tion and analysis of the model will be discussed. We recommend that some problems should be solved with effective computational help, although it is somewhat technologically intensive to do so. 1 Compound interest Let S ( t ) be the value of the investment at time t and r be the annual inter- est rate compounded continuously. Assume that the deposits or withdrawals take place at a constant rate k continuously, then dS dt = rS + k, (4) which by manipulation gives us S ( t ) = S (0) e rt + k r ( e rt - 1) . (5) Example 1. A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. Assuming that interest is compounded continuously and that the borrower makes payments contin- uously at a constant annual rate k , determine the payment rate k that is required to pay off the loan in 3 years. Also determine how much interest is paid during the 3-year period. Let us define S ( t ) to be the loan at time t , then dS dt = rS - k. (6) Thus by (5), S ( t ) = S (0) e rt - k r ( e rt - 1) . (7) We have S (0) = 8000 and S (3) = 0, therefore k = rS (0) e rt e rt - 1 = rS (0) 1 - e - rt = 10% × 8000 1 - e - 10% × 3 = 3086 . 64 / year . 2
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Now let us think about the second question, how to find the interest paid? We have the obvious fact that
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Tutorial3 - Applications: Modeling with First Order...

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