Thread: Differential Equation/Compound Interest problem. Help please?

I sorta need help with this problem. Work shown would be appreciated! And any other form of help =)

Suppose Ms. Lee is buying a new house and must borrow $200,000. 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 $6,000 extra to get the loan.) As an approximation, we assume that interest is compounded and payments are made continuously. Let:

M(t) = amount owed at time t (measured in years),
i = annual interest rate, and
p = annual payment.

Then the model for the amount owed is:
dM/dt = iM-p

a)How much does Ms. Lee pay in each case?
b)Which is a better deal over the entire time of the loan (assuming Ms. Lee does not invest money she would have paid in points)?
c)If Ms. Lee can invest the $6,000 she would have paid in points for the second mortgage at 5% compounded continuously, which is the better deal?

Thank you so much if you can help me through this problem =)

Hey Hiroi.

Try solving the DE first and then consider how each of the two different loans changes the values of i and p (they will be constants for each loan type).

Unfortunately...I wouldnt even know how to do that -_- haha ><

The differential equation is separable. Your independent variable is t and your dependent is p but you can get a dp/f(p) = dt*g(t) and integrate boths sides.

Hint: for this case f(p) = iM - p (use the other information to get g(t) and be aware that g(t) can be a constant independent of t as a special case).

Ohh I see, so when i separate it, should it come out to be:
dM/(iM-P)=dt?
Then I integrate both sides, leaving me with ln|iM-P|=t+c?
But from then on, what would I do?

Well, you are asked to find the amount paid over the life of the loan and the loan will be paid off when M(t)= 0. What is t when M= 0?

t will be 30 (30 year mortgage) when M = 0. But I don't understand how that helps with this problem? ><

If I can, I'd still like to ask for more help >< This problem has me completely stumped!

Recall that M = M(t) is a function of t so you need to get M explicitly in terms of t.

So when I solve for M, I got it to be:
M=((e^t+c)+P)/i
But what do I do with the C? How do I solve for it?

You usually have an initial condition and you re-arrange in terms of C = blah.