Example 3: Paying off a loan A similar situation is paying off a loan. Suppose that you took out college loans totaling $60, 000 with interest of 7.5%. You have an online payment plan which continuously deducts money from your bank account at a rate which comes out to $15, 000 per year. How long will it take you to pay off the loan? Again, this will be a rst-order linear differential equation. The change in the balance of the loan will go up as interest is accumulated, and go down as payments are made. As such, if 3:; represents the balance of the loan at time t, then the differential equation will have a positive term for the interest, and a negative term for the payments: y' = .0753: 15000 Again, the initial condition tells us that 11(0) = 60000. Using the techniques from class, we get the general solutlon POE) = 200000 + 0907\" and particular solution P0?) = 200000 1400006075\" In order to nd the amount of time it takes to pay off the loan, we need to solve for when the balance P(t) is zero. 0 = 200000 1400009073 200000 = 1400006075\" 200000 _ 9 _ 9075* 140000 _ 7 _