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Math Help - Differential equation help.

  1. #1
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    Differential equation help.

    Before I begin, I have to admit I'm hopeless at these things. Never gotten my head around them. Differentiation I'm okay with, but not these annoying buggers!

    Question:
    Consider the differential equation:
    2xy.dy/dx = y^2 - 9
    a. Find the steady state solutions (if any) and the general solution.
    b. Find the particular solution satisfying y = 5 when x = 4.

    Well, this is my attempt at answering these questions:
    rearrange the formula:
    dy.1/(y – 9) = dx.1/2xy
    integrate to dy and dx:
    ∫y/y – 9).dy = ∫1/2x.dx
    ln(y - 9) = ln(x)/2 + c
    ln(y - 9) = ln(x) + 2c
    e(ln(y - 9)) = e(ln(x) + 2c)
    y - 9 = Ax where A = e^2c
    y = Ax + 9

    when y = 5 and x = 4:
    25 = 4A + 9
    A = 4
    ln(A) = ln(4)
    c = ln(2)

    Is this correct? Personally I think not based on my previous attempts at doing these sort of questions!
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  2. #2
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    Its perrrrfect!

    Dont try to find c. Since the final equation is y^2 = 4x+9. A is enough.

    And moreover you need not ask anyone actually
    You can just substitute it back in the differential equation and do what you like doing.... that is differentiate and see if the equation holds
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  3. #3
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    Quote Originally Posted by Dr Zoidburg View Post
    Before I begin, I have to admit I'm hopeless at these things. Never gotten my head around them. Differentiation I'm okay with, but not these annoying buggers!

    Question:
    Consider the differential equation:
    2xy.dy/dx = y^2 - 9
    a. Find the steady state solutions (if any) and the general solution.
    b. Find the particular solution satisfying y = 5 when x = 4.

    Well, this is my attempt at answering these questions:
    rearrange the formula:
    dy.1/(y – 9) = dx.1/2xy
    integrate to dy and dx:
    ∫y/y – 9).dy = ∫1/2x.dx
    ln(y - 9) = ln(x)/2 + c
    ln(y - 9) = ln(x) + 2c
    e(ln(y - 9)) = e(ln(x) + 2c)
    y - 9 = Ax where A = e^2c
    y = Ax + 9

    when y = 5 and x = 4:
    25 = 4A + 9
    A = 4
    ln(A) = ln(4)
    c = ln(2)

    Is this correct? Personally I think not based on my previous attempts at doing these sort of questions!
    Yea of little faith. Your answer to (b) is correct. Applying your OK skill with differentiation would have shown you that:

    Differentiate your solution implicitly: 2y \, \frac{dy}{dx} = A \Rightarrow 2xy \, \frac{dy}{dx} = Ax.

    Substitute into the DE:

    Ax = (Ax + 9) - 9.

    Left hand side = right hand side.
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  4. #4
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    OMG!
    I got it right?!
    whoah...maybe it is all starting to sink in!
    thanks for the reassurance. That's my assignment finished, tg
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  5. #5
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    Quote Originally Posted by Dr Zoidburg View Post
    OMG!
    I got it right?!
    whoah...maybe it is all starting to sink in!
    thanks for the reassurance. That's my assignment finished, tg
    You found the steady state solutions, right?
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  6. #6
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    D'oh!
    Any help there would be appreciated!
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  7. #7
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    Quote Originally Posted by Dr Zoidburg View Post
    D'oh!
    Any help there would be appreciated!
    So .... what is a stationary solution? How do you find them? How does that apply to the given DE?
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  8. #8
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    Have I got this right:
    dy/dx = h(y).f(x) making
    h(y) = (y^2 - 9)/y
    setting h(y)=0 we get
    y = {-3,+3}

    These are the steady state solutions. correct?
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  9. #9
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    Quote Originally Posted by Dr Zoidburg View Post
    Have I got this right:
    dy/dx = h(y).f(x) making
    h(y) = (y^2 - 9)/y
    setting h(y)=0 we get
    y = {-3,+3}

    These are the steady state solutions. correct?
    Yes.
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