# Differential Equation

Suppose Ms. Lee is buying a new house and must borrow 150,000. She wants a 30-year mortgage and she has two choices. She can either borrow money at 7% per year with no points, or she can borrow the money at 6.5% per year with a charge of 3 points. (A "point" is a fee of 1% of the loan amount that the borrower pays the lender at the beginning of the loan. For example, a mortgage with 3 points requires Ms. Lee to pay 4,500 extra to get the loan.) As an approximation, we assume that interest is compounded and payments are made continuously. Let

$$M(t) = \text{amount owed at time } t\ \left(\text{measured in years}\right)$$ $$r= \text{annual interest rate, and}$$ $$p= \text{annual payment}$$

Then the model for the amount owed is

$$\frac{dM}{dt}=rM-p$$

Q.How much does Ms Lee has to pay in each case?

I have tried solving the DE, and i get
$$M(t)=C_1e^{rt} + \frac{p}{r}$$

Now what to do?

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