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Math Help - Help with Differential Eqn

  1. #1
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    Help with Differential Eqn

    I've been tearing my hair out for the past two hours trying to figure out why I can't do a differential equation.

    Here is the problem and the solution:

    Help with Differential Eqn-math-prob.png


    The bottom equation is supposed to be solution, but I have no idea how they got that.

    Here is my work trying to solve it (sorry about legibility, had to use a mouse):

    Help with Differential Eqn-math.png

    The (kvo+g) seems to just pop out of nowhere, and I have no idea where that comes from. Can anyone help?

    EDIT: to be clear, the Vo is just the constant of integration
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by bigallah View Post
    I've been tearing my hair out for the past two hours trying to figure out why I can't do a differential equation.

    Here is the problem and the solution:

    Click image for larger version. 

Name:	math-prob.png 
Views:	48 
Size:	11.1 KB 
ID:	20654


    The bottom equation is supposed to be solution, but I have no idea how they got that.

    Here is my work trying to solve it (sorry about legibility, had to use a mouse):

    Click image for larger version. 

Name:	math.png 
Views:	12 
Size:	59.6 KB 
ID:	20655

    The (kvo+g) seems to just pop out of nowhere, and I have no idea where that comes from. Can anyone help?

    EDIT: to be clear, the Vo is just the constant of integration
    Seeing the question in its entirety would be helpful; you've given us just a part of the question and I'm sure the beef of the information is contained in the part you haven't shown us.

    So please post the whole question.

    P.S.: This is more suitable for the differential equations subforum, not the calculus subforum. So post future DE questions in the Differential Equations subforum. Consider this a friendly warning.
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  3. #3
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    Problem is a part of 2-9)b) that can be found here:

    Instructor's Solutions Manual - Marion, Thornton - Classical Dynamics of Particles and Systems, 5th Ed!!!!!!!!!!

    I posted in the Calc section since this DiffEq is supposed to be stupid simple -- I remember learning how to solve 1st order ODEs in Calc rather than DiffEQ. You can keep it in whatever section you want, I just need an answer.
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  4. #4
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    Your solution is \frac{C}{k}e^{-kt}- \frac{g}{k}
    and the given solution is \frac{kv_0+ g}{k}e^{-kt}- \frac{g}{k}
    so the only difference is that your constant of integration, C, is replace by their kv_0+ g.

    I have not looked at the entire problem but I imagine there is some other, initial, condition that gives C= kv_0+ g
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  5. #5
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    \displaystyle m\,\frac{dv}{dt} = -mg - kmv

    \displaystyle \frac{dv}{dt} = -g - kv

    \displaystyle \frac{dv}{dt} + kv = -g.


    This is first order linear, so use an Integrating Factor \displaystyle = e^{\int{k\,dt}} = e^{kt}.

    Multiplying both sides by the Integrating Factor gives

    \displaystyle e^{kt}\,\frac{dv}{dt} + k\,e^{kt}v = -g\,e^{kt}

    \displaystyle \frac{d}{dt}\left(e^{kt}v\right) = -g\,e^{kt}

    \displaystyle e^{kt}v = \int{-g\,e^{kt}\,dt}

    \displaystyle e^{kt}v = -\frac{g\,e^{kt}}{k} + C

    \displaystyle v = -\frac{g}{k} + C\,e^{-kt}.


    They would have used some initial or boundary conditions to find \displaystyle C.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    Your solution is \frac{C}{k}e^{-kt}- \frac{g}{k}
    and the given solution is \frac{kv_0+ g}{k}e^{-kt}- \frac{g}{k}
    so the only difference is that your constant of integration, C, is replace by their kv_0+ g.

    I have not looked at the entire problem but I imagine there is some other, initial, condition that gives C= kv_0+ g
    Wow, I see it now. Thanks. If you take the \frac{kv_0+ g}{k}e^{-kt}- \frac{g}{k} equation and give it the conditions v(t=0)=vo then it works out to what I want. Lol, I guess I just made a bad assumption when trying to follow the solution.

    Many thanks to everyone who helped out!
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