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Math Help - Differential Equations, Some Integral Solutions

  1. #1
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    Differential Equations, Some Integral Solutions

    I have a vague memory of going over these types of integrals, but have yet to fully visualize and comprehend the patterns that they imply. I'm now stuck on a problem because the answer is different from mine and the solution guide skips it. Any help you can give to help me with the concept and/or the problem would be appreciated.

    What I think I understand about differential equations:
    A differential equation is an equation that relates a formula and it's derivative in the same equation. So that often means that a differential equation is in the for of dy/dx=f(y) where y=f(x), right? I've understood direction fields of differential equations and can piece together their pattern of divergence versus convergence and to what value they are diverging or converging to.

    My Conceptual Questions:
    1. Does the relationship between a differential equation and a direction field mean that a differential equation dy/dx=f(y) is always a pattern that tends to generalize the behavior of many different functions y=f(x)?

    2. Does every y=f(x) have at least one dy/dx=f(y) that could describe it? If so, are there ways to determine the later from the former? If not, what sorts of situations tend to call for the later?

    3. On a direction field, when a curve made from the field reaches a value y, is it's curve from that point onward identical (super imposable?) with any other curve drawn from the direction field from that same value of y onwards?

    The Problem, My answer, and the answer at the end of the book:
    Consider the field mice population equation in Ex1 [An equation for the population of field mice trends with a given owl population preying on them] dp/dt=0.5p-450. Find the time at which the population becomes extinct if p(0)=850. [t is in months]

    dp/dt=0.5p-450
    divide both sides by 1/2
    2dp/dt=p-900
    2dp=(p-900)dt
    integrate both sides
    2p+C=pt-900t+C
    2p-pt=-900t+C
    p=(-900t+C)/(2-t)
    p(0)=850=C/2
    C=1700
    0=(-900t+1700)/(2-t)
    2-t=-900t+1700
    899t=1698
    t=1.888765295months

    Book Answer
    t=2ln18 approx=5.78 months

    Any and all help is appreciated
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  2. #2
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    Quote Originally Posted by youngidealist View Post
    I have a vague memory of going over these types of integrals, but have yet to fully visualize and comprehend the patterns that they imply. I'm now stuck on a problem because the answer is different from mine and the solution guide skips it. Any help you can give to help me with the concept and/or the problem would be appreciated.

    What I think I understand about differential equations:
    A differential equation is an equation that relates a formula and it's derivative in the same equation. So that often means that a differential equation is in the for of dy/dx=f(y) where y=f(x), right? I've understood direction fields of differential equations and can piece together their pattern of divergence versus convergence and to what value they are diverging or converging to.

    My Conceptual Questions:
    1. Does the relationship between a differential equation and a direction field mean that a differential equation dy/dx=f(y) is always a pattern that tends to generalize the behavior of many different functions y=f(x)?

    2. Does every y=f(x) have at least one dy/dx=f(y) that could describe it? If so, are there ways to determine the later from the former? If not, what sorts of situations tend to call for the later?

    3. On a direction field, when a curve made from the field reaches a value y, is it's curve from that point onward identical (super imposable?) with any other curve drawn from the direction field from that same value of y onwards?

    The Problem, My answer, and the answer at the end of the book:
    Consider the field mice population equation in Ex1 [An equation for the population of field mice trends with a given owl population preying on them] dp/dt=0.5p-450. Find the time at which the population becomes extinct if p(0)=850. [t is in months]

    dp/dt=0.5p-450
    divide both sides by 1/2
    2dp/dt=p-900
    2dp=(p-900)dt
    integrate both sides
    2p+C=pt-900t+C
    2p-pt=-900t+C
    p=(-900t+C)/(2-t)
    p(0)=850=C/2
    C=1700
    0=(-900t+1700)/(2-t)
    2-t=-900t+1700
    899t=1698
    t=1.888765295months

    Book Answer
    t=2ln18 approx=5.78 months

    Any and all help is appreciated
    \frac{dp}{dt} = \frac{p}{2} - 450

    \frac{dp}{dt} = \frac{p - 900}{2}

    \frac{dt}{dp} = \frac{2}{p - 900}

    t = \int{\frac{2}{p - 900}\,dp}

    t = 2\ln{|p - 900|} + C


    Since when t = 0, p = 850

    0 = 2\ln{|850 - 900|} + C

    C = -2\ln{|-50|}

    C = -2\ln{50}.


    So t = 2\ln{|p - 900|} - 2\ln{50}

    t = 2\ln{\frac{|p - 900|}{50}}.


    You want to solve for t when p = 0.

    So t = 2\ln{\frac{|0 - 900|}{50}}

    t = 2\ln{\frac{900}{50}}

    t = 2\ln{18}.
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  3. #3
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    Thanks . The conceptual questions are still on my mind right now if anyone can still help me with them. At least now I can move forward.
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