Science:Math Exam Resources/Courses/MATH105/April 2011/Question 05 (a)

MATH105 April 2011

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Question 05 (a)

You open a savings account under the Escalator Plan with an initial deposit of $\$P$. The advantage of this plan is that the interest rate is not fixed, but grows proportionally with time as long as the account is alive. In other words, the money in the account collects interest at the annual rate of $rt$ at time $t,$ compounded continuously (here $r$ is a constant). You also keep adding money to the account in the form of a continuous deposit of $\$S(t)$ at time $t.$ Write down the initial value problem for the amount $A(t)$ in your account in time $t.$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Proceed step by step to create that differential equation: what is the influence of the continuous compounding? how do you take into account that you continuously add to your investment? how do we model the fact that the interest rate grows proportionally with time?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. Let $P$ be the initial deposit in the account and $A(t)$ be the amount of money in the account at time $t$. We denote the interest rate by $i$. The fact that we compound interest rate continuously translates into the differential equation ${\frac {dA}{dt}}=i\cdot A$ In addition, at every time we add the amount $S(t)$ to the account. Hence, the differential equation changes to ${\frac {dA}{dt}}=i\cdot A+S$ Furthermore, the interest rate grows proportionally with time. This means that $i=rt$ for some constant $r$. Thus, we arrive at ${\frac {dA}{dt}}=rt\cdot A+S$ We also know that $A(0)=P$ because $P$ is the initial deposit.

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