Question: You decide to start a 401k account. You begin with an initial investment of $5000, and you continuously add to it at a rate of $1000 per year. At the same time, the account accrues interest at a continuous rate of 5% annually. Let V (t) be the value of the account t years after starting it.
(i) Set up a differential equation which describes dV/dt in terms of V .
(ii) This is a separable differential equation. Solve it to get a general formula for V (t). Be sure to keep the necessary constant around.
(iii) Use the initial value to find the constant and get a precise formula for V (t). (iv) When will the 401k be worth $50,000?