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$\frac{dr}{E_{r}}=\frac{rd\theta}{E_{\theta}}$

$\varphi=\frac{qa^{2}}{r^{3}}\left(3\cos^{2}\theta-1\right)$

$E_{r}=-\frac{\partial\varphi}{\partial r}=3\frac{qa^{2}}{r^{4}}\left(3\cos^{2}\theta-1\right)$

$E_{\theta}=-\frac{\partial\varphi}{r\partial\theta}=6\frac{qa^{2}}{r^{4}}\sin\theta\cos\theta$

$2\frac{dr}{r}=\frac{(3\cos^{2}\theta-1)d\theta}{\sin\theta\cos\theta}=\frac{3\cos\theta d\theta}{\sin\theta}-\frac{d\theta}{\sin\theta\cos\theta}$

$\int\frac{3\cos\theta d\theta}{\sin\theta}=\int\frac{3d\sin\theta}{\sin\theta}=3\ln|\sin\theta|+C_{1}$

$\int\frac{d\theta}{\sin\theta\cos\theta}=\int\frac{\cos\theta d\theta}{\sin\theta\cos^{2}\theta}=\int\frac{d\text{tg}\theta}{\text{tg}\theta}=\ln|\text{tg}\theta|+C_{2}$

$3\ln|\sin\theta|+C_{1}-\ln|\text{tg}\theta|-C_{2}=\ln\left(\sin^{2}\theta|\cos\theta|\right)+C'_{1}$

$2\frac{dr}{r}=2\ln r+C_{3}$

$r=C\sqrt{\sin^{2}\theta|\cos\theta|}$