How do you convert $y=x-2y+xy^2$ into a polar equation?

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Answer 1 · 2018-05-21T16:06:17

$r=(costheta-sintheta)/((2-costheta)sinthetacostheta)$

$y=x-2y+xy^2$
$x=rcostheta$
$y=rsintheta$
Thus,
$rsintheta=rcostheta-2rcosthetaxxrsintheta+rcosthetaxx(rsintheta)^2$

$rsintheta=rcostheta-2r^2sinthetacostheta+r^2sin^2thetacostheta$

$rsintheta-rcostheta=r^2(-2sinthetacostheta+sin^2thetacostheta)$

$r(sintheta-costheta)=r^2sinthetacostheta(-2+costheta)$

$r(sintheta-costheta)+r^2sinthetacostheta(2-costheta)$

$r(sintheta-costheta+rsinthetacostheta(2-costheta))=0$

$r=0$

$sintheta-costheta+rsinthetacostheta(2-costheta)=0$

$rsinthetacostheta(2-costheta)=costheta-sintheta$
$r=(costheta-sintheta)/((2-costheta)sinthetacostheta)$

How do you convert y=x-2y+xy^2 y=x-2y+xy^2 into a polar equation?