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Thread: [SOLVED] differential equation

  1. #1
    Super Member malaygoel's Avatar
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    [SOLVED] differential equation

    Solve
    $\displaystyle (x^2 + y^2 + x)dx=-xydy$


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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by malaygoel View Post
    Solve
    $\displaystyle (x^2 + y^2 + x)dx=-xydy$


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    Malay
    recall that exact differential equations are those of the form $\displaystyle M(x,y)~dx + N(x,y)~dy = 0$, where $\displaystyle \frac {\partial M}{\partial y} = M_y = N_x = \frac {\partial N}{\partial x}$

    So this guy starts off looking like an exact equation, right, with $\displaystyle M(x,y) = x^2 + y^2 + x$ and $\displaystyle N(x,y) = xy$

    but lo and behold, $\displaystyle M_y = 2y \ne N_x = y$. thus we need to find an integrating factor

    Recall that we can find the integrating factor $\displaystyle \mu$ by the differential equation:

    $\displaystyle \frac {d \mu}{dx} = \frac {M_y - N_x}{N} \mu$

    so here we have:

    $\displaystyle \frac {d \mu}{dx} = \frac {2y - y}{xy} \mu = \frac {\mu}{x}$

    $\displaystyle \Rightarrow \mu (x) = x$

    multiplying through by $\displaystyle x$ we obtain:

    $\displaystyle \left( x^3 + xy^2 + x^2 \right)~dx + \left( x^2y \right)~dy = 0$

    which is now an exact equation, since $\displaystyle M_y = 2xy = N_x$

    can you continue?
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