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Math Help - Separating Variables

  1. #1
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    Separating Variables

    Hi everyone...

    Im stuck on this question (and its xmas eve i dont want to be spending hours on this what appears to be a very simple problem....aaah! )

    so in my lecture notes, i have the equation,

    dy/dt = y+1/t-1 given the boundary condition y=1 at t=0

    separating variables gives,

    integral( dy/y+1) = integral (dt/y-1)
    ln(y+1) = ln(t-1) + lnc - c

    OR (now this is where im lost..how did they get c - bc and how on earth did c equal -2??? you'll see...)

    y+1/t-1 = c - bc tells us that c = -2

    y = 2(1-t)-1
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  2. #2
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    Quote Originally Posted by matlabnoob View Post

    integral( dy/y+1) = integral (dt/y-1)
    ln(y+1) = ln(t-1) + lnc - c

    You could say

    \ln(y+1) = \ln(t-1) + c

    \ln(y+1) = \ln(t-1) + \ln(c)

    \ln(y+1) = \ln(c(t-1))

    y+1 = c(t-1)

    y = c(t-1)-1

    Now use your I.C. y=1 at t=0 to find c.

    You should find c = -2

    Quote Originally Posted by matlabnoob View Post


    OR (now this is where im lost..how did they get c - bc and how on earth did c equal -2??? you'll see...)
    Where did b come from?
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  3. #3
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    \frac{dy}{dt}=\frac{y+1}{t-1}

    Separate:

    \frac{dy}{y+1}=\frac{dt}{t-1}

    Integrate:

    ln(y+1)=ln(t-1)+C

    e to both sides:

    y+1=e^{ln(t-1)+C}=e^{C}(t-1)

    But e^{C} is a constant we can rename and call C_{1}

    y=C_{1}(t-1)-1

    Now, use your IC to solve for C1
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  4. #4
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    Re: Separating Variables

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