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Math Help - Word problem with DE

  1. #1
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    Word problem with DE

    Ok, I simply cannot get the answer that they have in the solutions manual, nor do they explain how they got the answer, so I really have no clue what I'm doing wrong.

    The instructions for the problem are: Investment. A brokerage firm opens a new real estate investment plan for which the earnings are equivalent to continuous compounding at the rate of r. The firm estimates that deposits from investors will create a net cash flow of Pt dollars, where t is the time in years. The rate of change in the total investment A is modeled by

    \frac{dA}{dt}=rA+Pt

    (a) Solve the differential equation and find the total investment A as a function of t. Assume that A = 0 when t = 0.

    (b) Find the total investment A after 10 years given that P = $500,000 and r = 9%.

    Ok, so I rearranged the DE to be A' - rA = Pt and then determined that  P(t) = -r and Q(t)=Pt. From there, I found IF=e^{\int-rdt}=e^{-rt}.

    I then mutiplied each side by the IF to get e^{-rt}A'-rAe^{-rt}=Pte^{-rt}.

    From there I doublechecked that the IF was correct, it was, and then I did \int \frac{dA}{dt} (e^{-rt} * A) = \int Pte^{-rt}

    The integral on the left cancels with the derivative so

     e^{-rt}A=\int Pte^{-rt}

    I then pulled P out as a constant to get e^{-rt}A=P\int te^{-rt}

    I then did parts, using u=t, du=dt, V=e^{-rt} to get

     e^{-rt}A=P(te^{-rt} - \int e^{-rt}dt)

    From there I used u=-rt and du=-rdt and since I only have a dt I did -\frac{1}{r}du=dt

    Solving the final integral I came up with

    e^{-rt}A=P(te^{-rt}+\frac{1}{r}e^{-rt}+C)

    Dividing both sides by  e^{-rt} to get A by itself, I got

    A=P(t+\frac{1}{r}+Ce^{rt})

    Which is different from what the solutions manual got, which was

     A=\frac{P}{r^2}(-rt-1+Ce^{rt})

    What am I doing wrong?
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  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
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    Quote Originally Posted by emttim84 View Post
    Ok, I simply cannot get the answer that they have in the solutions manual, nor do they explain how they got the answer, so I really have no clue what I'm doing wrong.

    The instructions for the problem are: Investment. A brokerage firm opens a new real estate investment plan for which the earnings are equivalent to continuous compounding at the rate of r. The firm estimates that deposits from investors will create a net cash flow of Pt dollars, where t is the time in years. The rate of change in the total investment A is modeled by

    \frac{dA}{dt}=rA+Pt

    (a) Solve the differential equation and find the total investment A as a function of t. Assume that A = 0 when t = 0.

    (b) Find the total investment A after 10 years given that P = $500,000 and r = 9%.

    Ok, so I rearranged the DE to be A' - rA = Pt and then determined that  P(t) = -r and Q(t)=Pt. From there, I found IF=e^{\int-rdt}=e^{-rt}.

    I then mutiplied each side by the IF to get e^{-rt}A'-rAe^{-rt}=Pte^{-rt}.

    From there I doublechecked that the IF was correct, it was, and then I did \int \frac{dA}{dt} (e^{-rt} * A) = \int Pte^{-rt}

    The integral on the left cancels with the derivative so

     e^{-rt}A=\int Pte^{-rt}

    I then pulled P out as a constant to get e^{-rt}A=P\int te^{-rt}

    I then did parts, using u=t, du=dt, V=e^{-rt} to get

     e^{-rt}A=P(te^{-rt} - \int e^{-rt}dt)

    From there I used u=-rt and du=-rdt and since I only have a dt I did -\frac{1}{r}du=dt

    Solving the final integral I came up with

    e^{-rt}A=P(te^{-rt}+\frac{1}{r}e^{-rt}+C)

    Dividing both sides by  e^{-rt} to get A by itself, I got

    A=P(t+\frac{1}{r}+Ce^{rt})

    Which is different from what the solutions manual got, which was

     A=\frac{P}{r^2}(-rt-1+Ce^{rt})

    What am I doing wrong?
    Is P a function or a variable or a constant? if it is a constant this is a simple seperable differential equation
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Yuma, AZ, USA
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    your integration by parts is incorrect
    the correct integration is

    \int Pte^{-rt}=P\left[ -\frac{t}{r}e^{-rt}-\frac{1}{r^2}e^{-rt}+C \right]
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  4. #4
    Junior Member
    Joined
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    Quote Originally Posted by TheEmptySet View Post
    your integration by parts is incorrect
    the correct integration is

    \int Pte^{-rt}=P\left[ -\frac{t}{r}e^{-rt}-\frac{1}{r^2}e^{-rt}+C \right]
    Right, I figured the integration is incorrect, but where is the error so I can see how to get the correct answer, and not just have the correct answer?
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