Problem 19 | Separation of Variables

$dr = b(\cos \theta \, dr + r \sin \theta \, d\theta)$

$dr = b\cos \theta \, dr + br \sin \theta \, d\theta$

$dr - b\cos \theta \, dr = br \sin \theta \, d\theta$

$(1 - b\cos \theta) \, dr = br \sin \theta \, d\theta$

$\dfrac{(1 - b\cos \theta) \, dr}{r(1 - b\cos \theta)} = \dfrac{br \sin \theta \, d\theta}{r(1 - b\cos \theta)}$

$\dfrac{dr}{r} = \dfrac{b \sin \theta \, d\theta}{1 - b\cos \theta}$

$\displaystyle \int \dfrac{dr}{r} = \int \dfrac{b \sin \theta \, d\theta}{1 - b\cos \theta}$

$\ln r = \ln (1 - b\cos \theta) + \ln c$

$\ln r = \ln c(1 - b\cos \theta)$

$r = c(1 - b\cos \theta)$           answer