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Separable Differential Equation

  1. Apr 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve the separable differential equation
    \frac{dx}{dt} = \frac{6}{x} ,
    and find the particular solution satisfying the initial condition
    x(0) = 7.

    x(t) = .

    2. Relevant equations

    \[ \frac{dy}{dt} = ky \]

    3. The attempt at a solution

    lnx=6

    x=e^6
     
  2. jcsd
  3. Apr 5, 2009 #2
    I can't interpret you attempt at LaTeX (I can't do it either), but somewhere in your integration you lost the time dependence. That is essentially what has gone wrong in your solution.
     
  4. Apr 5, 2009 #3
    Dr. D is correct, as it looks like you forgot to actually perform a separation of variables. To do this, you get all the x's and dx's to one side and all the t's and dt's two one side. Then you integrate both sides, and after some simplifying, you will use your initial condition to solve for your integration constant.

    The relevant equation you posted is not the same here, because you have x in the denominator, not the numerator.

    To do [itex]\LaTeX[/itex] on the Physics Forums, you enclose your [itex]\LaTeX[/itex] within come code brackets. For example:
    [tex] \frac{dx}{dt} = \frac{6}{x} [/tex]
    Click on the graphic to see the code. Use itex if you want your graphics to fit on the same line as your text.
     
  5. Apr 5, 2009 #4
    Solve the separable differential equation
    dx/dy = 6/x ,
    and find the particular solution satisfying the initial condition
    x(0) = 7.
     
  6. Apr 5, 2009 #5
    Yes. We know what to do, but you must show us some work. You just posted what you have already posted, except switching your t to a y. Did you try what I mentioned? Do you know how to perform separation of variables?

    Follow my original advice. If you get stuck, show us what work you've done or tried, and then we'll go from there.
     
  7. Apr 5, 2009 #6
    [tex]\intx/6[/tex]

    = 6*ln(x)

    0=6ln(7)

    0=11.6755

    x(t)=6ln(t)+11.6755

    I have no idea where I am going wrong
     
  8. Apr 5, 2009 #7
    Right here! The integral of x/6 is not 6*ln(x). I don't understand the rest of your equations.

    I will get you started:
    [tex]
    \begin{align*}
    \frac{dx}{dt} &= \frac{6}{x} \\
    x \,dx &= 6 \,dt \tag{\text{the separation of variables}} \\
    \int x \,dx &= \int 6 \,dt \tag{\text{integrate both sides}}
    \end{align*}
    [/tex]

    Now see if you can finish finding the solution. Remember that whatever you get as a solution, then you should check that it actually is a solution. If it doesn't satisfy the differential equation and initial condition you were given, then it's not a solution.

    You should know that the differential equation
    [tex]\frac{dx}{dt} = \frac{6}{x}[/tex]
    means that the derivative of some function x(t) is 6/x(t). We're solving for that function x(t) (a solution to the differential equation), which is a function of t.
     
  9. Apr 6, 2009 #8
    Look really hard at the line that n!kofeyn has labeled (the separation of variables). This seems to have been the step that was eluding you, and it is nothing more than algebra!
     
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