Differential Equation Question

Question

Let y(t) represent your retirement account balance, in dollars, after t years. Each year the account earns 4% interest, and you deposit 4% of your annual income. Your current annual income is $38000, but it is growing at a continuous rate of 2% per year.

Write the differential equation modeling this situation.

Answer 1

Albee,

 

y′(t) represents the rate of change of the retirement account balance after t years.

 

The change due to interest is 4% of the balance, y(t), or .04y.

 

The change due to depositing 4% of your annual income is .04 A(t), where A(t) is the annual income after t years. Since this is growing at a continuous rate of 2% per year, it satisfies the differential equation A′(t) = .04 A(t), A(0) = 38000. This equation describes exponential growth and has solution A(t) = 38000e^.04t. Thus, the change due to the income deposit is .04 times this, or 1520e^.04t.

 

Putting all of this together, you have the DE:

 

y′(t) = .04y + 1520e^.04t.

 

This can be rewritten in the form of a first-order linear DE:

 

y′ - .04y = 1520e^.04t.

 

This should be what you need to answer the question. I am sorry it took so long for someone to send a reply. i just noticed this today.