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Math Help - Last question of the day on differential equations

  1. #1
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    Last question of the day on differential equations

    So I have this problem which states use separation of variables to find the general solution of the differential equation.

    \frac{dy}{dx}=\sqrt\frac{x}{y} so to start, I'll move the dx to the other side to get:

    dy=\sqrt\frac{x}{y}dx

    Now, to get rid of the square root, can I simply square both sides to get:

    d^{2}y=\frac{x}{y}dx^2 then divide by y to get:

    \frac{1}{y}d^{2}y=xdx^{2}

    And then from there, integrate both sides twice? The only issue I see is after integrating once, I get:

    lnydy=\frac{1}{2}x^2dx+c

    And I don't quite remember how to integrate a natural log...
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  2. #2
    Super Member wingless's Avatar
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    \frac{dy}{dx}=\sqrt{\frac{x}{y}}

    \frac{dy}{dx}=\frac{\sqrt{x}}{\sqrt{y}}

    \sqrt{y}~dy = \sqrt{x}~dx

    Now integrate..
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by emttim84 View Post
    So I have this problem which states use separation of variables to find the general solution of the differential equation.

    \frac{dy}{dx}=\sqrt\frac{x}{y} so to start, I'll move the dx to the other side to get:

    dy=\sqrt\frac{x}{y}dx

    Now, to get rid of the square root, can I simply square both sides to get:

    d^{2}y=\frac{x}{y}dx^2 then divide by y to get:

    \frac{1}{y}d^{2}y=xdx^{2}

    And then from there, integrate both sides twice? The only issue I see is after integrating once, I get:

    lnydy=\frac{1}{2}x^2dx+c

    And I don't quite remember how to integrate a natural log...
    And just for you knowledge using integration by parts \int\ln(ax)dx=x\cdot{ln(ax)}-\int{\frac{x\cdot{a}}{a\cdot{x}}}=x\cdot{ln(ax)}-x+C
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  4. #4
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    ....well, don't I feel like a retard. And now you know why I'm not a math major. :P thanks guys.
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by emttim84 View Post
    ....well, don't I feel like a retard. And now you know why I'm not a math major. :P thanks guys.
    Haha that is why we are here!
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