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{\bf Question}

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Determine whether the differential equation $$\displaystyle x \sin
t \frac{dx}{dt} + t \sin x = 0$$

is exact, and, if so, find the general solution.

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{\bf Answer}

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Consider $\displaystyle x \sin t \frac{dx}{dt} + t \sin x = 0$

$\displaystyle \frac{\partial}{\partial t}[x \sin t] = x \cos t$
\hspace{.5in} $\displaystyle \frac{\partial}{\partial x}[t \sin x]
= t \cos x$

$\displaystyle x \cos t \not= t \cos x$ therefore equation is NOT
exact.



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