Write the differential equation modeling this situation

Question

Let y(t) represent your retirement account balance, in dollars, after t years. Each year the account earns 8% interest, and you deposit 8% of your annual income. Your current annual income is $42000, but it is growing at a continuous rate of 2% per year.

Write the differential equation modeling this situation.

dy/dt = 

Answer 1

Hi Gabe,

Thanks for your question!

Our differential equation should take the form

dy(t)/dt = (Increase in $ due to Interest) + (Increase in $ due to Deposit),

where

Increase in $ due to Interest = (Interest Rate)*(Account Balance) = 0.08y(t),

Increase in $ due to Deposit = (Percent of Annual Income)*(Annual Income) = 0.08a(t).

We're told that the annual income a(t) initially is $42000, meaning a(0) = 42000. This income is changing continuously at an annual rate of 2% percent per year, meaning

da(t)/dt = 0.02a(t).

We can solve this differential equation to find a(t) = Cexp(0.02t), where C is an arbitrary constant. Given a(0) = 42000, we find C = 42000, so that

a(t) = 42000exp(0.02t).

Substituting this a(t) back into our expression for the amount of money we're depositing annually into the retirement account, we find that our differential equation for y(t) is

dy(t)/dt = 0.08y(t) + 0.08*42000exp(0.02t).