Question

Mr. Brown has taken out a loan of value $1,000 with a 5% APR. He has been paying off the loan at a constant rate of r = $150/year. Set up a linear ODE for the balance of the loan P(t) as a function of time t in years. [If you're having trouble getting started, please see Example 2 from

Answer 1

`(dp)/(dt)` = 0.05P - 150

Explanation:

Let P(t) be the balance of the loan at time t years,

Let P(t) will satisfy
`(dp)/(dt)=rP-R`

where r = annual interest rate

R = per year payment rate

R = $150/year

r =
`(5)/(100)` = 0.05
`(5)/(100)`

Now we have the following picture,

Interest Balance Payment

5% ⇒ P(t) ⇒ $150/year

(0.05)

Therefore, a linear ODE satisfied by P(t) is given by

`(dp)/(dt)` = 0.05P - 150