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Deriving a differential equation for a loan/interest problem

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  1. Sep 29, 2011 #1
    1. The problem statement, all variables and given/known data
    You borrow money from a friend at a continuous interest rate of r% per month. You want to pay your friend back as quickly as you can at the beginning, but reduce your payment rate over time. You decide to pay off at a continuously decreasing rate given by K₀e[itex]^{-at}[/itex], in dollars per month.

    Write a differential equation that describes how much you owe and solve it.


    2. Relevant equations
    None


    3. The attempt at a solution
    Let S be the amount borrowed -

    dS/dt = rS - K₀e[itex]^{-at}[/itex]

    S' - rS = -K₀e[itex]^{-at}[/itex]

    S'(I(x)) - rS(I(x)) = -K₀e[itex]^{-at}[/itex](I(x))

    Se[itex]^{-rt}[/itex] = -K₀[itex]\int[/itex]e[itex]^{-t(a+r)}[/itex]

    Se[itex]^{-rt}[/itex] = ...

    This is where I get stuck, I have don't understand how to integrate -K₀[itex]\int[/itex]e[itex]^{-t(a+r)}[/itex], any hints?
     
  2. jcsd
  3. Sep 29, 2011 #2

    rock.freak667

    User Avatar
    Homework Helper

    if 'a' and 'r' are constants then you can simply recall that ∫ ekt = (1/k)ekt+ constant.
     
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