I don't know if this is appropriate to ask so please let me know if I should keep this post or not.
While calculating some accelerated trajectories, i encountered a rather tedious differential equation of the form, $$ \frac{da^0}{d\tau} + \frac{\Big(\mu - \frac{GQ^2}{r} \Big) }{GQ^2 + r^2 - 2r\mu} (u^0 a^1 + u^1 a^0) = \vert a \vert^2 u^0$$ where, $u^i$ is a 4-vector of velocity (dependent on $\tau$), $G$, $Q$ can be treated as constants along with $\mu$ and $r$ is a variable. Here, $a^i$ is the acceleration 4-vector and the expression for which is,
$$a^0 = \frac{du^0}{d\tau} + \frac{2\Big(\mu - \frac{GQ^2}{r} \Big) }{GQ^2 + r^2 - 2r\mu} (u^1 u^0)\\ a^1 = \frac{du^1}{d\tau} - \frac {(Mr-Q^{2})}{r^{2}-2Mr+Q^{2}}(u^0)^2 + \frac {Q^{2}-Mr}{Q^{2}r-2Mr^{2}+r^{3}} (u^1)^2$$ So, after combining these equations (where $\vert a \vert^2$ is also a constant ) and substituting the later expressions into the former equation,we get a long differential equation like,
$$\frac{d}{d\tau} \Big(\frac{du^0}{d\tau} + \frac{2\Big(\mu - \frac{GQ^2}{r} \Big) }{GQ^2 + r^2 - 2r\mu} (u^1 u^0)\Big) + \frac{\Big(\mu - \frac{GQ^2}{r} \Big) }{GQ^2 + r^2 - 2r\mu} \Bigg(u^0 \Big( \frac{du^1}{d\tau} - \frac {(Mr-Q^{2})}{r^{2}-2Mr+Q^{2}}(u^0)^2 + \frac {Q^{2}-Mr}{Q^{2}r-2Mr^{2}+r^{3}} (u^1)^2\Big) + u^1 \Big(\frac{du^0}{d\tau} + \frac{2\Big(\mu - \frac{GQ^2}{r} \Big) }{GQ^2 + r^2 - 2r\mu} (u^1 u^0)\Big)\Bigg) = \vert a \vert^2 u^0 $$
which I am not getting anywhere with. Any help/clue would be greatly appreciated. Thanks!
EDIT: I think the additional information about the $a^1$ equation could be needed. $$ \frac{da^1}{d\tau} = \vert a \vert^2 u^1$$