Science:Math Exam Resources/Courses/MATH105/April 2011/Question 05 (a)/Solution 1
Let be the initial deposit in the account and be the amount of money in the account at time . We denote the interest rate by . The fact that we compound interest rate continuously translates into the differential equation
In addition, at every time we add the amount to the account. Hence, the differential equation changes to
Furthermore, the interest rate grows proportionally with time. This means that for some constant . Thus, we arrive at
We also know that because is the initial deposit.