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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?

  • 1
    What is the s in $\frac{s}{r}$? Now you know what M($0$) should be, right? It is the amount owed at the beginning, which should give you a value for $C_1$, so that you have a formula for $M(t)$. If I understood the problem well, the payments will be made over 30 years. If this is the case, the formula M(t) for t=30 and r=$7%$% and for M(t) for t=30 and $6.5%$%+ $4,500 should give you different values, which you can compare to each other.2011-07-01
  • 0
    thanks, Gary. $\frac{s}{r}$ should actually be $$\frac{p}{r}$$. As for $M(0)$ is the amount she owed at the beginning of the time, it's an unknown. we only know $M(30)$ which is equal to $0$.2011-07-01
  • 0
    Why do you assume continuous compounding and payments? I've always seen this kind of problem handled by difference equations (recurrence relations), rather than differential equations.2011-07-01
  • 0
    @Gerry: Maybe you don't want to use continuous compounding, but the bank does. Or maybe a writer wants to use it in his differential equations textbook.2011-07-01
  • 0
    With the exception of consumer and small business loans, continuous compounding is standard.2011-07-01
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    @user6312, really? I thought continuous compounding didn't exist outside of math texts. The interest is increasing while you're standing in line at the bank to pay it!2011-07-02
  • 0
    @Gerry Myerson: Yes, that's why they have so few tellers. More seriously, it really is the norm, for example, in interbank transactions. But being practically minded, they round billion dollar transactions to the nearest cent.2011-07-02

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