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I need to solve this differential equation. YWhat I'm looking for is a way to simplify this equation. Can anybody give me hints/tricks to understand the following equation better:

$$xy'(\cos(2x^3)\cos(x)-\sin(2x^3)\sin(x)) - $$ $$y(6x^3+x)\sin(2x^3+x)$$ $$+(12x^3+2x)\sin(2x^3+x)=0$$

Ultimately I want to seperate variables and integrate to solve for $y(x)$

Thanks for your help :)

Edit: Please If you can help me solve this differential equation, any help would be appreciated :)

thanks

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2 Answers 2

up vote 1 down vote accepted

Note that : $\cos(2x^3)\cos(x)−\sin(2x^3)\sin(x) = \cos(2x^3+x)$

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It seems to me that two first term of the equation: $$xy'(\cos(2x^3)\cos(x)-\sin(2x^3)\sin(x)) - y(6x^3+x)\sin(2x^3+x)$$ is somehow a part of total differential of $$xy(\cos(2x^3+x))$$ Since $$d\left(x\cos(2x^3+x)y\right)=x\cos(2x^3+x)y'+y\left(\cos(2x^3+x)-(6x^3+x)\sin(2x^3+x)\right)$$

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@amWhy: I become refreshed whenever you wish me that Amy. Thanks :-) –  B. S. Apr 6 '13 at 6:50

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