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Math Help - Past papers with brief solutions :(

  1. #1
    Newbie VerdictGuilty's Avatar
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    Question Past papers with brief solutions :(

    Ive been going through exam papers and theres a question on differential equations which I cant get to grips with

    m*dv/dt = mg-kv [g> k/mv]

    dv/dt= g-kv/m

    dt = g-kv/m dv

    im not sure on how to integrate this can anyone help explain it?


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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Past papers with brief solutions :(

    This is a linear ODE, and you want to write it in standard form:

    \frac{dv}{dt}+\frac{k}{m}v=g

    Now, calculate the integrating factor \mu(t)=e^{\int\frac{k}{m}\,dt}=e^{\frac{k}{m}t} and multiply the ODE by this:

    e^{\frac{k}{m}t}\frac{dv}{dt}+\frac{k}{m}e^{\frac{  k}{m}t}v=ge^{\frac{k}{m}t}

    Now, the left side is the differentiation of a product:

    \frac{d}{dt}\left(e^{\frac{k}{m}t}v \right)=ge^{\frac{k}{m}t}

    Can you proceed from here?
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  3. #3
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    Re: Past papers with brief solutions :(

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    Re: Past papers with brief solutions :(

    The integration for dt is simply "t".
    The expression on the RHS is being integrated with respect to "v" so you should treat everything else as a constant and then proceed.
    The expression ∫g-kv/m dv = g∫dv- k/m∫vdv
    = gv- k/m(v^2/2)+c

    Hope this helps.
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  5. #5
    Newbie VerdictGuilty's Avatar
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    Re: Past papers with brief solutions :(

    hey thanks that helps alot guys I should be able to go from here thank you
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