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Thread: Exact Equation

  1. #1
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    Exact Equation

    $\displaystyle (e^x \sin{y} -3x^2) \,dx +(e^x \cos{y} + \frac{y^{-2/3}}{3}) \, dy = 0$

    $\displaystyle (e^x \sin{y} -3x^2) +(e^x \cos{y} + \frac{y^{-2/3}}{3}) \frac{dy}{dx} = 0
    $

    $\displaystyle M = e^x \sin{y} -3x^2 \,, N = e^x \cos{y} + \frac{y^{-2/3}}{3}$

    $\displaystyle \frac{\partial M}{\partial y} = e^x \cos{y}$

    $\displaystyle \frac{\partial N}{\partial x} = e^x \cos{y}$

    Since
    $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$,
    there exists an F such that
    $\displaystyle \frac{d}{dx}F(x, y) = 0$

    $\displaystyle F = \int{M dx}$
    $\displaystyle F = \int{e^x \sin{y} -3x^2 dx} = e^x \sin{y} -x^3 + h(y)$

    $\displaystyle \frac{\partial F}{\partial y} = N $

    $\displaystyle e^x \cos{y} + \frac{y^{-2/3}}{3} = e^x \sin{y} -x^3 + h'(y)$

    I can't find any way to get past this point. Is there any possible function $\displaystyle h'(y)$ that would satisfy this equality? Or have I made an error?
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  2. #2
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    Your working is a little off...

    Since $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

    that means

    $\displaystyle \frac{\partial F}{\partial x} = M$ and $\displaystyle \frac{\partial F}{\partial y} = N$.

    So

    $\displaystyle \frac{\partial F}{\partial x} = e^x\sin{y} - 3x^2$

    $\displaystyle F = \int{e^{x}\sin{y} - 3x^2\,dx}$

    $\displaystyle = e^x\sin{y} - x^3 + g(y)$.


    $\displaystyle \frac{\partial F}{\partial y} = e^x\cos{y} + \frac{y^{-\frac{2}{3}}}{3}$

    $\displaystyle F = \int{e^x\cos{y} + \frac{y^{-\frac{2}{3}}}{3}\,dy}$

    $\displaystyle = e^x\sin{y} + y^{\frac{1}{3}} + h(x)$.


    So we have at the same time...

    $\displaystyle F = e^x\sin{y} - x^3 + g(y)$ and $\displaystyle F = e^x\sin{y} + y^{\frac{1}{3}} + h(x)$.

    This means $\displaystyle g(y) = y^{\frac{1}{3}} + C_1$ and $\displaystyle h(x) = -x^3 + C_2$.


    Putting everything together gives

    $\displaystyle F = e^x\sin{y} - x^3 + y^{\frac{1}{3}} + C$ where $\displaystyle C = C_1 + C_2$.
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  3. #3
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