Results 1 to 2 of 2

Thread: Investment word problem w/differential equations

  1. #1
    Junior Member
    Joined
    Nov 2007
    Posts
    53

    Investment word problem w/differential equations

    Ok, here's the problem:

    Investment. Let A(t) be the amount in a fund earning interest at the annual rate of r, compounded continuously. If a continuous cash flow of P dollars per year is withdrawn from the fund, then the rate of decrease of A is given by the differential equation $\displaystyle \frac{dA}{dt}=rA-P$

    where $\displaystyle A=A_0$ when $\displaystyle t=0$

    (a) Solve this equation for A as a function of t.

    (b) Use the result of part (a) to find A when $\displaystyle A_0$=$2,000,000, r=7%, P=$250,000, and t=5 years

    (c) Find $\displaystyle A_0$ if a retired person wants a continuous cash flow of $40,000 per year for 20 years. Assume that the person's investment will earn 8%, compounded continuously.


    Ok, so C is simple enough once A and B is done since you basically just plug in $\displaystyle t=20, r=0.08, A=$40,000$ for the variables. Part A & B is what's getting me. Here's what I did.

    I rearranged it into a first-order linear DE so $\displaystyle A' - rA = -P$ and my $\displaystyle P(x)=-r$ so the IF is $\displaystyle e^{\int-rdt}=e^{-rt}$

    Then I multiplied the original equation on both sides by the IF to get

    $\displaystyle e^{-rt}(A'-rA)=-Pe^{-rt}$

    From there...

    $\displaystyle \int \frac{d}{dt}(e^{-rt}*A) = \int (-Pe^{-rt})dt$

    $\displaystyle e^{-rt}A= \frac{P}{r}e^{-rt} + C$

    $\displaystyle A= \frac{P}{r} + Ce^{rt}$

    What's tripping me up is $\displaystyle A=A_0$ only when $\displaystyle t=0$ so I'm not sure how to proceed.

    I can set $\displaystyle t=0$ and plug in $\displaystyle A_0$ for $\displaystyle A$ with the above general soln to get $\displaystyle A_0= \frac{P}{r} + C$

    but then part B asks me to input t as one of the variables and $\displaystyle t$ doesn't show up in this new soln so I'm not sure where to go from here.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Nov 2007
    Posts
    53
    Can anyone help me with this please? I only need help with the $\displaystyle A_0$ part, not the entire part B & C.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Math Word Problems: Investment
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Aug 30th 2009, 08:20 PM
  2. Differential Equations Word Problem...
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Mar 12th 2009, 05:20 AM
  3. Replies: 4
    Last Post: Apr 29th 2008, 08:08 PM
  4. Investment word problem
    Posted in the Business Math Forum
    Replies: 2
    Last Post: Mar 13th 2007, 12:28 PM
  5. Investment word problem
    Posted in the Business Math Forum
    Replies: 1
    Last Post: Mar 13th 2007, 10:03 AM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum