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Math Help - How do i deal with this separable differential equation?

  1. #1
    Member Jones's Avatar
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    How do i deal with this separable differential equation?

    Hello,

    I need to solve this separable differential equation

    4y^3\frac{dy}{dx}=x^{-2}~~~~~,y(1) = 1

    We have 4y^3~dy = \frac{1}{x^2}dx

    y^4 = -\frac{1}{x}

    But what is \bigg(-\frac{1}{x}\bigg)^{1/4} ?
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by Jones View Post
    Hello,

    I need to solve this separable differential equation

    4y^3\frac{dy}{dx}=x^{-2}~~~~~,y(1) = 1

    We have 4y^3~dy = \frac{1}{x^2}dx

    y^4 = -\frac{1}{x}

    But what is \bigg(-\frac{1}{x}\bigg)^{1/4} ?
    The correct answer is y^4 = - \frac{1}{x} + C. Then you substitute y(1) = 1 to get the value of C. Then you make y the subject (if that's what the question expects).
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  3. #3
    Member Jones's Avatar
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    This is confused,

    Don't you want the answer in terms of y?
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  4. #4
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    e^(i*pi)'s Avatar
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    You want y to be the subject usually. ie: y = f(x).

    Find C using the way Mr F said: 1 = -\frac{1}{1}+C

    Once the value of C is known you can take the 4th root (obviously substitute in the value of C calculated in the previous step) : y = \sqrt[4]{-\frac{1}{x}+C}


    As you said you cannot take the 4th root of a negative number so you will have domain restrictions in your answer since -\frac{1}{x} +C \geq 0
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