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RideAnS [48]
3 years ago
15

Find Integrating Fector: ylogy dx + ( x-logy )dy=0

Mathematics
1 answer:
12345 [234]3 years ago
4 0
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

so that it is now linear in x. An integrating factor would

\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
x\log y=\displaystyle\int\frac{\log y}y\,\mathrm dy
x\log y=\dfrac12\log^2y+C
x=\dfrac12\log y+\dfrac C{\log y}
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