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Thread: differential equation w/ initial condition

  1. #1
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    differential equation w/ initial condition

    Solve the differential equation with given conditions

    (dy/dt)=(1/2)y

    y(2)=100
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  2. #2
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    Quote Originally Posted by thedoge View Post
    Solve the differential equation with given conditions

    (dy/dt)=(1/2)y

    y(2)=100
    $\displaystyle y'=\frac{1}{2}y$
    Divide by $\displaystyle y$.
    First check if $\displaystyle y=0$ is a solution (it is).
    Now look for other solutions $\displaystyle y\not = 0$,
    $\displaystyle \frac{y'}{y}=\frac{1}{2}$
    Integrate,
    $\displaystyle \int \frac{y'}{y} dx = \int\frac{1}{2} dx$
    Thus,
    $\displaystyle \ln |y|=\frac{1}{2}x+C'$
    Thus,$\displaystyle e^{\ln |y|}=e^{\frac{1}{2}x+C'}$
    $\displaystyle |y|=Ce^{\frac{1}{2}x},C>0$
    $\displaystyle y=Ce^{\frac{1}{2}x}$
    $\displaystyle 100=Ce^{\frac{1}{2}(2)}$
    $\displaystyle 100=Ce^{1}=Ce$
    $\displaystyle C=100/e=100e^{-1}$.
    Thus,
    $\displaystyle y=Ce^{\frac{1}{2}x}=100e^{-1}e^{\frac{1}{2}x}=100e^{\frac{1}{2}x-1}$
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  3. #3
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    As fast and amazing as usual PH.

    If you don't mind me asking, do you have a specific profession outside this forum? You seem to be a master of mathematics which should open up about any occupation to you
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  4. #4
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    Quote Originally Posted by thedoge View Post

    If you don't mind me asking, do you have a specific profession outside this forum?
    Yes! I am a fashion designer.
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  5. #5
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    Haha. A comedian too.

    Unless you're serious. One can never know online;]
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  6. #6
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    Hrm. I follow everything you said perfectly and even worked it out that way myself, but apparently this problem needs to evaluate to a number.

    What is the value of x?

    Nevermind. I see what the problem was.

    *** for future reader's reference replace the x in 100*e^(.5x-1) with a t
    Last edited by thedoge; Feb 4th 2007 at 08:11 PM.
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