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Math Help - Help with Differential Equation

  1. #1
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    Help with Differential Equation

    Hello all,

    I'm trying to solve a differential equation derived from Boyle's Gas Law which is setup as y' = f(x,x',y) and I can't figure out how to start.

    The equation is below:

    dV/dt = -V/P * dP/dt

    I'm really lost on this one because of the differential of two different variables.

    Any information you guys could provide would be really helpful.

    Thanks,
    Mitch
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  2. #2
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    Jester's Avatar
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    Re: Help with Differential Equation

    You can write the ODE as

    \dfrac{1}{V} \dfrac{dV}{dt} + \dfrac{1}{P} \dfrac{dP}{dt} = 0

    \dfrac{d \ln V}{dt} +  \dfrac{d \ln P}{dt} = 0

    or

    \dfrac{d (\ln V + \ln P) }{dt} = 0 or more conveniently \dfrac{d \ln PV  }{dt} = 0.

    This can now be integrated
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  3. #3
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    Re: Help with Differential Equation

    What I would do is write \frac{dV}{dt}= -\frac{V}{P}\frac{dP}{dt} in 'differential form':
    dV= -\frac{V}{P}dP and then as

    \frac{dV}{V}= -\frac{dP}{P}
    and integrate both sides.

    Of course, this gives exactly the same answer as Jester's suggestion.
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