A Car Loan: Suppose the amount of principal left to be paid back on a car loan is given by P(t) and suppose a payment of k dollars is made on the loan each month. Then the rate at which the remaining principal P(t) changes with respect to time (in $/month) is the net result of the monthly payment of k dollars (which is a positive constant) and the interest which is proportional to the remaining principal (here the proportionality constant is the monthly continuous interest rate r).

a. Write a differential equation to represent this situation using the function and constants named above.

b. A student purchases a car for $18,000 with a 4-year loan at 3% annual interest, compounded continuously. Calculate the monthly interest rate r and then use this value to rewrite the DE from part a. Then solve this differential equation for P(t). Be sure to determine the value of C and state your model for P(t) clearly.

c. Use your model for P(t) to find the monthly payment k so that the loan in paid off (P(t) = 0) in 4 years (48 months). Give your answer rounded to the nearest cent.

d. Derive the general payment formula, solving for k in terms of r, n (the number of months), and P0 (the initial principal), showing all work to support it.

This is a practice question for an ODE exam tomorrow so could you please be very detailed and write every step out.

3rd time posting and so far this is all I have gotten from the Chegg Experts below...please take your time on it and go through every step for A(which is below and assuming it is correct), B, C, and D. Thanks in advance!

a) is P' = rP-kt here t is in months k is the amount of the emi , so k needs to be multiplied by t as kt will be the amount paid after t monts and the remaining amount of P(t) would be pr => P' = Pr - kt

b) nothing

c) nothing

d) nothing