01
Design your traditional IRA
You are planning a traditional IRA for retirement: contributions during your working life are pre-tax, and withdrawals during retirement are taxed according to your income bracket when withdrawing. Your plan is to start the fund with $20,000 when you turn 30, and then to contribute a fixed yearly amount (modeled with a continuous yearly addition rate) each year until you turn 65. You invest the IRA in a broad market index with average yearly returns of 10% (not adjusted for inflation). Starting at age 65 you stop contributing and start withdrawing $70,000 per year after taxes for the next 25 years of retirement, and you would like to have $200,000 remaining in your estate at age 90, both figures measured in today’s dollars. Your tax bracket with this retirement income is 20%. Inflation is assumed to be a steady 2.5% per year. Everything is compounded continuously. What should you choose for the yearly contributions ?
02
Bacteria growth on surface of spherical clump
A spherical clump of bacteria is growing in such a way that only the bacteria on its surface can reproduce, because only these are exposed to new food sources in the environment, while these surface bacteria do reproduce with exponential growth. Recall the formulas for volume and surface area of a sphere:
Find a differential equation satisfied by the number of bacteria as a function of time.
03
Studying an autonomous equation
Sketch the phase line of the following ODE:
Classify each equilibrium trajectory as stable or unstable. Draw some representative solution trajectories using the qualitative information of your phase line.
04
Gompertz equation
Tumor growth is sometimes modeled by the Gompertz equation:
What are the equilibrium lines? Are they stable or unstable? What is the inflection point on a trajectory from the lower to the upper equilibrium?
Now solve the equation explicitly.