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Math Help - 1st order ODE, Reduction to Separable Form Help required

  1. #1
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    1st order ODE, Reduction to Separable Form Help required

    Hey guys,
    I'm very new to this form of solving differential equations and I've come to a stump in a problem.

    The problem is simply to find the general solution to y'=(4x^2+y^2)/(xy)

    I have a guide that also shows the steps to the solution, but the area I am stuck on is identical to the guide, where i don't see how it got from one step to the next.

    This is what I have so far;

    Set  u=\frac{y}{x}, y=xu, y'=u+xu'

    y'=\frac{4x^2}{xy}+\frac{y^2}{xy}

    y'=\frac{4x}{y}+\frac{y}{x}

    u+xu'=\frac{4}{u}+u

    Therefore;

    xu'=\frac{4}{u}

    Now, at this point I assumed it would be very easy to separate the variables to;

    uu'=\frac{4}{x}

    And solve by integration. However, the guide goes from

    xu'=\frac{4}{u}

    to

    2u du = \frac{8}{x} dx
    My question is... what step am i missing that u becomes 2u and 4/x becomes 8/x?

    Thanks for your time
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  2. #2
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    Re: 1st order ODE, Reduction to Separable Form Help required

    Quote Originally Posted by Nuingaer View Post
    Hey guys,
    I'm very new to this form of solving differential equations and I've come to a stump in a problem.

    The problem is simply to find the general solution to y'=(4x^2+y^2)/(xy)

    I have a guide that also shows the steps to the solution, but the area I am stuck on is identical to the guide, where i don't see how it got from one step to the next.

    This is what I have so far;

    Set  u=\frac{y}{x}, y=xu, y'=u+xu'

    y'=\frac{4x^2}{xy}+\frac{y^2}{xy}

    y'=\frac{4x}{y}+\frac{y}{x}

    u+xu'=\frac{4}{u}+u

    Therefore;

    xu'=\frac{4}{u}

    Now, at this point I assumed it would be very easy to separate the variables to;

    uu'=\frac{4}{x}

    And solve by integration. However, the guide goes from

    xu'=\frac{4}{u}

    to

    2u du = \frac{8}{x} dx
    My question is... what step am i missing that u becomes 2u and 4/x becomes 8/x?

    Thanks for your time
    They multiplied both sides by 2. Why, I don't know. But it's valid...
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  3. #3
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    Re: 1st order ODE, Reduction to Separable Form Help required

    A simpler method :
    Attached Thumbnails Attached Thumbnails 1st order ODE, Reduction to Separable Form Help required-simpler-method.jpg  
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