 Solution

With the apsidal origin chosen as specified, $G=0$. For $p_{\varphi }\ne 0$, rearrangement givesLet $u=1/r$. Since $du/d\varphi =-p_r/p_{\varphi }$, the equation is $u^{\prime }=-(u-mk/p_{\varphi }^2)\tan \varphi$. Its solution, continued through points where the intermediate division fails, isAs printed, $e$ has units of inverse length and is not the usual dimensionless eccentricity. Writing $\ell =p_{\varphi }^2/(mk)$ gives the standard form $r=\ell /(1+\epsilon \cos \varphi )$ with $\epsilon =e\ell$. The orbit is an ellipse for $|\epsilon |<1$, parabolic for unit magnitude and hyperbolic for magnitude greater than one on its allowed angular interval. The purely radial case $p_{\varphi }=0$ must be handled separately and is excluded by these divisions.