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Math Help - solving diff equation w/ bournoulli's equation

  1. #1
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    solving diff equation w/ bournoulli's equation

    I am trying to find the general solution of
    y(dy/dx) + x = sqrt(x^2 + y^2)<br />

    I'm having issues figuring out what n should be, as I can't get the y's together to figure out the powers. How would I do this.

    thanks
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  2. #2
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    Quote Originally Posted by snoboarder2k6 View Post
    I am trying to find the general solution of
    y(dy/dx) + x = sqrt(x^2 + y^2)<br />

    I'm having issues figuring out what n should be, as I can't get the y's together to figure out the powers. How would I do this.

    thanks
    y\frac{dy}{dx} + x = \sqrt{x^2 + y^2}


    \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \frac{\sqrt{x^2 + y^2}}{y}


    \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \sqrt{\frac{x^2 + y^2}{y^2}}


    \Rightarrow \frac{dy}{dx} + \frac{x}{y} = \sqrt{\left( \frac{x}{y} \right)^2 + 1}


    The standard technique is to now make the substitution y = vx \Rightarrow \frac{dy}{dx} = v + \frac{dv}{dx}. After some simplifying you get


    \frac{dv}{dx} + 1 + v^2 = \sqrt{1 + v^2}.

    This DE is seperable, leading to x = \frac{\sqrt{v^2 + 1} + 1}{v} + C. Now substitute back y = vx \Rightarrow v = \frac{y}{x} .....
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  3. #3
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    ah, tricky, I forgot that's how to get something into a sgrt. thanks
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