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Math Help - Bernoulli's Equation problem

  1. #1
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    Bernoulli's Equation problem

    problem:
    6ydx - x(2x+y)dy = 0

    n = ?
    P = ?
    Q = ?
    V = ?

    general solution = ?
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  2. #2
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    Quote Originally Posted by cazimi View Post
    problem:
    6ydx - x(2x+y)dy = 0

    n = ?
    P = ?
    Q = ?
    V = ?

    general solution = ?
    You first need to re-write this equation in Bernoulli form

    \frac{dy}{dx} + P(x)y = Q(x) y^n. Isolating \frac{dy}{dx} gives

    \frac{dy}{dx} = \frac{6y^2}{x(2x^3+y)}

    which isn't of the right form but isolating \frac{dx}{dy}

    gives

    \frac{dx}{dy} - \frac{x}{6y} = \frac{x^4}{3y^2} where  P = -\frac{1}{6y}, \;\;\ Q = \frac{1}{3y^2} \,\,\, n = 4 . I don't know what your V is?

    Before we discuss the solution, do you know how to solve Bernoulli equations?
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  3. #3
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    Yes, I was referring to the V as

    <br />
v=exp(\int\P,dx)
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  4. #4
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    Quote Originally Posted by cazimi View Post
    Yes, I was referring to the V as

    <br />
v=exp(\int\P,dx)
    Before you find your integrating factor, you will need to linearize the equation

    so

    \frac{dx}{dy} - \frac{x}{6y} = \frac{x^4}{3y^2}

    becomes

     \frac{1}{x^4} \frac{dx}{dy} - \frac{1}{x^3} \frac{1}{6y} = \frac{1}{3y^2}

    and then let u = \frac{1}{x^3}. See how you go from here.
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