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Math Help - Differential Equation/Compound Interest problem. Help please?

  1. #1
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    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 =)
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  2. #2
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    Re: Differential Equation/Compound Interest problem. Help please?

    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).
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  3. #3
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    Re: Differential Equation/Compound Interest problem. Help please?

    Unfortunately...I wouldnt even know how to do that -_- haha ><
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  4. #4
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    Re: Differential Equation/Compound Interest problem. Help please?

    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).
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  5. #5
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    Re: Differential Equation/Compound Interest problem. Help please?

    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?
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  6. #6
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    Re: Differential Equation/Compound Interest problem. Help please?

    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?
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  7. #7
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    Re: Differential Equation/Compound Interest problem. Help please?

    t will be 30 (30 year mortgage) when M = 0. But I don't understand how that helps with this problem? ><
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  8. #8
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    Re: Differential Equation/Compound Interest problem. Help please?

    If I can, I'd still like to ask for more help >< This problem has me completely stumped!
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  9. #9
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    Re: Differential Equation/Compound Interest problem. Help please?

    Recall that M = M(t) is a function of t so you need to get M explicitly in terms of t.
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  10. #10
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    Re: Differential Equation/Compound Interest problem. Help please?

    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?
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  11. #11
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    Re: Differential Equation/Compound Interest problem. Help please?

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