Chapter D1.4, Problem 37E

A bad loan Consider a loan repayment plan described by the initial value problem

     $\begin{array}{cc}{B}^{\prime }(t)=0.03B-600,& B(0)=40,000,\end{array}$

 where the amount borrowed is B(0) = $40,000, the monthly payments are $600, and B(t) is the unpaid balance in the loan.

a.    Find the solution of the initial value problem and explain why B is an increasing function.

b.    What is the most that you can borrow under the terms of this loan without going further into debt each month?

c.    Now consider the more general loan repayment plan described by the initial value problem

     $\begin{array}{cc}{B}^{\prime }(t)=rB-m,& B(0)=B_0,\end{array}$

 where r > 0 reflects the interest rate, m > 0 is the monthly payment, and B₀ > 0 is the amount borrowed. In terms of m and r, what is the maximum amount B₀ that can be borrowed without going further into debt each month?