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Math Help - help me solve this

  1. #1
    Junior Member ceode's Avatar
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    help me solve this

    SOLVE
    (x*y^2)dx - e^(1/(x^3))dx - (x^2)*y dy = 0
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  2. #2
    A Plied Mathematician
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    Try u(x)=y^{2}(x).
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  3. #3
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    Jester's Avatar
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    Noting that d\left( \dfrac{y^2}{x^2}\right) = \dfrac{2x^2dy - 2 x y^2 dx}{x^4} might help.
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  4. #4
    Super Member General's Avatar
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    xy^2dx-yx^2dy-e^{x^{-3}}dx=0

    Multiply the equation by 2:

    2xy^2dx-2yx^2dy-2e^{x^{-3}}dx=0

    Recall that d\left(\left(\dfrac{x}{y}\right)^2 \right)=d\left(\dfrac{x^2}{y^2}\right)=\dfrac{2xy^  2dx-2yx^2dy}{y^4} \implies  y^4 \, d\left(\left(\dfrac{x}{y}\right)^2 \right)=2xy^2dx-2yx^2dy

    The equation will be:

    y^4 \, d\left(\left(\dfrac{x}{y}\right)^2 \right) - 2 e^{x^{-3}}dx=0

    Devide by x^4

    \left(\left(\dfrac{x}{y}\right)^{-2}\right)^2 d\left(\left(\dfrac{x}{y}\right)^2 \right) - 2 x^{-4} e^{x^{-3}}dx = 0

    Let t=\left(\dfrac{x}{y}\right)^2

    The equation will be:

    \dfrac{1}{t^2} dt - 2 x^{-4} e^{x^{-3}}dx=0

    Now, Integrate.
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  5. #5
    Junior Member ceode's Avatar
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    Quote Originally Posted by General View Post
    xy^2dx-yx^2dy-e^{x^{-3}}dx=0

    Multiply the equation by 2:

    2xy^2dx-2yx^2dy-2e^{x^{-3}}dx=0

    Recall that d\left(\left(\dfrac{x}{y}\right)^2 \right)=d\left(\dfrac{x^2}{y^2}\right)=\dfrac{2xy^  2dx-2yx^2dy}{y^4} \implies  y^4 \, d\left(\left(\dfrac{x}{y}\right)^2 \right)=2xy^2dx-2yx^2dy

    The equation will be:

    y^4 \, d\left(\left(\dfrac{x}{y}\right)^2 \right) - 2 e^{x^{-3}}dx=0

    Devide by x^4

    \left(\left(\dfrac{x}{y}\right)^{-2}\right)^2 d\left(\left(\dfrac{x}{y}\right)^2 \right) - 2 x^{-4} e^{x^{-3}}dx = 0

    Let t=\left(\dfrac{x}{y}\right)^2

    The equation will be:

    \dfrac{1}{t^2} dt - 2 x^{-4} e^{x^{-3}}dx=0

    Now, Integrate.


    Thank you
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