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Musya8 [376]
3 years ago
8

How is 77,184 the area of South Dakota

Mathematics
1 answer:
ivolga24 [154]3 years ago
8 0
The area of the state of South Dakota is 77,184 square miles.
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Rewrite the ODE as

y\log y\,\mathrm dx+(x-\log y)\,\mathrm dy=0\iff y\log y\dfrac{\mathrm dx}{\mathrm dy}+x=\log y
\dfrac{\mathrm dx}{\mathrm dy}=\dfrac1{y\log y}x=\dfrac1y

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\mu(y)=\exp\left(\displaystyle\int\frac{\mathrm dy}{y\log y}\right)=e^{\log(\log y)}=\log y

Multiply both sides by \mu(y) to get

\log y\dfrac{\mathrm dx}{\mathrm dy}=\dfrac1yx=\dfrac{\log y}y
\dfrac{\mathrm d}{\mathrm dy}[x\log y]=\dfrac{\log y}y
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