Science:Math Exam Resources/Courses/MATH105/April 2010/Question 05 (a)

MATH105 April 2010

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[hide] Question 05 (a)
A person purchased a home at the price of $ 300,000, paid a down payment equal to 20% of the purchase price, and financed the remaining balance with a 25 year term mortgage. Assume that the person makes payments continuously at a constant annual rate $A$ and that the interest is compounded continuously at the rate of 5%. Write down the differential equation that is satisfied by the amount $y(t)$ of money owed on the mortgage at time $t$ .

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[show] Hint
How can we use a differential equation to model the rate of the amount of money owed, ${\frac {{\textrm {d}}y}{{\textrm {d}}t}}$ ? What are the factors that increase the amount of money we owe? What are the factors that decrease it?

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[show] Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. At any given time, two things are happening: the owed money is growing at a rate of 5% and the amount of money is going down at a rate of A. Therefore, we write: ${\frac {dy}{dt}}=0.05y-A$ . As an initial condition, we impose that $y(0)=0.8\times 300,000=240,000$ i.e. the amount owed is eighty percent of the purchase price since twenty percent was initially paid.