Science:Math Exam Resources/Courses/MATH105/April 2011/Question 05 (a)/Solution 1

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.