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Math Help - Can any one solve This with intregrated factor method

  1. #1
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    Can any one solve This with intregrated factor method

    Please solve this with Intregrating factor method
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  2. #2
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    I assume that you already have taken both partial derivatives.

    When we want to find a suitable integrating factor, sometimes it's not so easy, so we may introduce the following cases.

    CASE I. Integrating factor dependent of x. Suppose that

    \frac{{\dfrac{{\partial M}}{{\partial y}} - \dfrac{{\partial N}}<br />
{{\partial x}}}}<br />
{N}

    it's a function that depends only of x, which we'll denote it by g(x). Then, an integrating factor for the given equation is \mu(x)=\exp\int g(x)\,dx.

    CASE II. Integrating factor dependent of y. If we have that

    \frac{{\dfrac{{\partial N}}{{\partial x}} - \dfrac{{\partial M}}{{\partial y}}}}{M}

    it's only a function of y, denoted by h(y), then \mu(y)=\exp\int h(y)\,dy is an integrating factor for the differential equation on the form M(x,y)\,dx+N(x,y)\,dy=0.

    CASE III. Integrating factors of the form x^my^n. If they exist m and n such that

    \frac{{\partial M}}{{\partial y}} - \frac{{\partial N}}{{\partial x}} = m\frac{N}{x} - n\frac{M}{y},

    then \mu(x,y)=x^my^n is an integrating factor for the differential equation mentioned previously.

    CASE IV. If they exist functions P(x) y Q(y) which satisfy

    \frac{{\partial M}}{{\partial y}} - \frac{{\partial N}}{{\partial x}} = N(x,y)P(x) - M(x,y)Q(y),

    then an integrating factor for the differential equation is \mu(x,y)=\exp\int P(x)\,dx\cdot\exp\int Q(y)\,dy.

    Note that the case IV includes to the cases I, II y III if we take Q(y)=0,\,P(x)=0 and P(x)=\frac mx,\,Q(y)=\frac ny; respectively.
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