Tutorial3

# Tutorial3 - Applications Modeling with First Order...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Now let us think about the second question, how to ﬁnd the interest paid? We have the obvious fact that

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 7

Tutorial3 - Applications Modeling with First Order...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online