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I was told to solve this without integration and to use implicit diffentiation.

$x^3 y^{\prime} - \dfrac{3y}{x} = x^3 e^{\left(x - \dfrac{1}{x^3}\right)}$

I am utterly lost, any suggestions.

I can get to

$ x^4y^{\prime} - 3y = x^4 e^{\left(x - \dfrac{1}{x^3}\right)}$

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    @MartinSleziak I wasn't complaining, very happy with your edit, much better than changing ( to \left(.2012-11-07

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Hint: Multiply both sides by the integrating factor $e^{1/x^3}$.

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    If you're going to solve a differential equation, there's no way to avoid integration (definite or indefinite) at some stage...2012-11-07