Mr. John want to buy a house and he must borrow $150,000 from the bank. He wants a 30-year mortgage and he has 2 choice.
Choice #1, he can borrow money at 7% per year with no point ( each point is 1% of the amount of the loan that Mr. John has to pay at the beginning of the loan).
Choice 2: he can borrow money at 6.5% per year with charge of 3 point ( this means he has to pay $4,500 at the beginning to get the loan).
Let the model of the amount owed is
$dM/dt=rM-p$
for $M(t)$ is the amount owed at time $t$,$r$ is the annual interest rate , and $p$ is annual payment
a) which deal is better over the entire time of the loan (assume Mr. John doesn't invest the money he would have paid in points)?
b) if Mr.John invest the $4,500 he would have paid in points for the second mortgage at 5% compound continuously, which is better deal?
This is not a differential equation, because 30-year mortgages are simply not compounded continuously. Using a geometrical series, one may derive that the total amount paid on a loan of principal $P$ at annual interest $i$, compounded monthly for $n$ years, is
$$M = \frac{P\, i\, n}{1-\left(1+\frac{i}{12}\right)^{-12 n}}$$
for the numbers above and $i=0.07$, the total paid out is $M \approx \$359,263$. For $i=0.065$, $m \approx \$341,317$, if the points are paid up front. If however, the points are rolled into the principal, the total amount paid is $\$351,556$. So using the points here is better even when rolled into the principal.
For the second part, if the debtor invests the $\$4,500$ at $5\%$ compounded continuously, then the debtor would have
$$4,500 \, e^{(0.05) 30} \approx \$20,167.60$$
which trumps the savings on the points even in the best-case scenario of paying the points up front.