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How do I do this?

Given $\csc{x \text{ } \frac{dy}{dx}}=xy$, express $y$ in terms of $x$

http://i.imgur.com/IXTXQ.png

I got $\frac{dy}{dx}=\frac{xy}{\csc{x}}$, then how can I remove the $y$? I am thinking maybe I need to use substitution or something?

UPDATE: Correct Working?

Is my working correct?

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    @jiewmeng $y=e^{-x\cos x+\sin x+C}=e^C\cdot e^{-x\cos x+\sin x}$ is the answer. No, $v$ dosesn't need it. Add the constant after you finish integrating by parts.2011-11-23

2 Answers 2

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HINT: The technique you need to use here is called "Separation of Variables". The idea is that for the differential equation $ \frac{dy}{dx} = f(x,y) $, in the special case where we can write $f$ as a product of functions of $x$ and $y$ separately, i.e $f(x,y) = h(x)g(y) $, then we write the equation in the form $ \frac{dy}{g(y) } = h(x) dx .$

Then we have only a single variable on each side, and we integrate both sides. So here, we should write $\int \frac{dy}{y} = \int x \sin x dx ,$ evaluate the integrals, and then solve for $y.$ The left integral is simply a natural log, while the integral on the right can be found by integration by parts.

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HINT: Rewrite it as $\frac{\mathrm{d} y}{y} = x \sin(x) \mathrm{d} x$, which is equivalent to $\int \frac{\mathrm{d} y}{y} = \int x \sin(x) \mathrm{d} x + C$. Can you evaluate these integrals ? Then use initial conditions to determined the integration constant $C$.