Question

A person starts, at time t = 0, to pay part of their salary continuously into an investment fund at a rate of P dollars per year. The fund earns interest at a rate r per year, compounded continuously. The amount of money in the fund after t years, M(t), then satisfies the differential equation

dM /dt =P+rM,
with M(0)=0.
(i) Solve the differential equation using the integrating factor method. (ii) If the person can pay in $1,250 each year, and the interest rate is 5%, how long will it take for the amount of money in the fund to reach $10,000?

Answer 1

The differential equation is solved using the integrating factor method, yielding an expression for the amount of money in the fund, M(t). Given the annual payment of $1,250 and a 5% interest rate, the time required to achieve a fund of $10,000 can be calculated by applying the derived formula.

Step-by-step explanation:

Solving the Differential Equation

To solve the differential equation dM/dt = P + rM, where t represents time, P the payment rate, and r the interest rate, we use the integrating factor method. We find the integrating factor μ(t) which is ert. By multiplying through by this integrating factor, the equation becomes μ(t)dM/dt = μ(t)P + rμ(t)M. This can be rewritten as d(μ(t)M)/dt = μ(t)P, which implies that μ(t)M is an antiderivative of μ(t)P. Integration of both sides yields μ(t)M = ∞(μ(t)P) + C, where C is the constant of integration. Applying the initial condition M(0) = 0, we get C = 0, so M(t) = (P/r)(ert - 1).

Calculating Time to Reach Goal

Given a payment of $1,250 per year, and an interest rate of 5%, we want to find how long it takes for the fund to reach $10,000. Using the previously found solution M(t) = (P/r)(ert - 1), we set M(t) to $10,000, P to 1,250, and r to 0.05, and solve for t. This requires finding the moment when (1,250/0.05)(e0.05t - 1) = 10,000. Solving for t gives us the time needed to reach the goal.