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Math Help - No idea how to find this numerical value from this first order differential equation

  1. #1
    s3a
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    No idea how to find this numerical value from this first order differential equation

    Here is the question (and Answer under "Differential equation solution"):
    "Find the solution of the differential equation that satisfies the given initial condition."
    dy/dx = x/y, y(0) = -3 - Wolfram|Alpha

    I'm ok when it involves only variables but when it gives me y(0) = -3, it throws me off.
    My work:

    dy/dx = x/y
    integral of ydy = integral of xdx
    1/2 * y^2 = 1/2 * x^2
    y = x

    That makes no sense.

    Can someone please explain to me what the question is really asking as well as help me solve it? I get confused because it gives me numbers and the answer is in variables.

    Thanks in advance!
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  2. #2
    Member Nacho's Avatar
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    you forget the integration constant, i mean, you must to have: y^2=x^2+k and whit y(0)=-3 you can obtain value of k
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  3. #3
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    \int y \, dy = \int x \, dx

    \implies \tfrac{1}{2}y^2+C_1 = \tfrac{1}{2}x^2+C_2

    You can just combine the two constants and get

    \implies \tfrac{1}{2}y^2 = \tfrac{1}{2}x^2+C

    Then you can simplify further and use the initial condition to solve for the constant.

    (This is basically repeating what Nacho said, but I just wanted to make it clear why there is only one constant instead of two.)
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  4. #4
    s3a
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    Ok so I get

    y = +/- sqrt(x^2 + 9)

    and the answer is
    y = -sqrt(x^2 + 9)

    my current question is...why do I choose the negative?
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  5. #5
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    Does the positive solution satisfy y(0) = -3?
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  6. #6
    s3a
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    I see. Thanks!
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