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{\bf Question}

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Classify, but do not solve, the equation $\displaystyle t^2
\frac{dx}{dt} + 2xt = 0$

under as many as possible of the following classifications:
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separable, $\frac{dx}{dt} = f\left( \frac{x}{t} \right)$, exact,
linear.
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{\bf Answer}

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$\displaystyle t^2 \frac{dx}{dt} + 2xt = 0$

$\displaystyle t^2\frac{dx}{dt} + 2xt = \frac{d}{dt}(xt62)
\Rightarrow \mathrm{exact}$
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No powers of $x^2$ or $\left( \frac{dx}{dt}\right)$ etc
$\Rightarrow$ linear.
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Equation can be rewritten as $\frac{dx}{dt} = -2\frac{x}{t}
\Rightarrow f(x,t) = f(\frac{x}{t}) = -2\frac{x}{t}$
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$$t^2\frac{dx}{dt} + 2xt = 0 \Rightarrow \frac{dx}{dt} =
(-2x)\left(\frac{1}{t}\right) = g(x)h(t) \Rightarrow {\rm
Separable}$$



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