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Math Help - How To Find The General Solution?

  1. #1
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    How To Find The General Solution?

    1. How do you find the general solution for the given differential equation?
    dP/dt = P(P+3)

    Answer: P = (3Ae^(3t))/(1-Ae^(3t))

    2. Of the pine trees in Manning Park, 400 000 are infested with a fungus. If aerial spraying with 400 kg/km^2 of a fundicide will reduce the number of trees infested to 40 000, how much is needed per square kilometre to reduce the number of trees infested to 2 000? (assume an exponential relationship.)

    Answer: 920.4 kg/km^2
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  2. #2
    MHF Contributor alexmahone's Avatar
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    \frac{dP}{dt}=P(P+3)

    \frac{dP}{P(P+3)}=dt

    \frac{dp}{P}-\frac{dp}{P+3}=3dt

    Integrating both sides,

    ln P-ln (P+3)=3t+C

    ln \frac{P}{P+3}=3t+C

    \frac{P}{P+3}=e^{3t}e^C

    \frac{P}{P+3}=Ae^{3t}

    P=PAe^{3t}+3Ae^{3t}

    P(1-Ae^{3t})=3Ae^{3t}

    P=\frac{3Ae^{3t}}{1-Ae^{3t}}
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