This is my first time asking a question here, so please tell me if I have made a mistake or if anything is unclear.
I am working on my high school research project on the motion of a ball falling through flowing water and derived eq. 1.
$b$ is the height of the ball in the situation. This equation was derived by adding the effects of gravity, buoyancy, and drag forces.
I want to use the equation to find or approximate the value of $b$ in terms of $t$, given the values of other constants.
Eq. 1: $$\frac{db}{dt}=\int_{0}^{t}{-g}\,dt+\int_{t_1}^{t}\left[\frac{3g\rho_w}{4r^3\rho_i}\left[\left(h_2-b+r\right)^2r-\frac{1}{3}\left(h_2-b+r\right)\right]+\frac{3r\rho_w}{16\rho_i}\left(\frac{db}{dt}-\frac{1}{2}k^2t-\frac{1}{2}k\sqrt{h_0}\right)^2\right]dt$$
$b, h_2$ are function of $t$, time. Everything else $(g, \rho_w, \rho_i, r, k, h_0, t_1)$ is constant.
However, $h_2$ in the formula is a function of $t$ and is given the following formula which includes $b$ :
Eq.2 $$h_2=\frac{A_c+\frac{1}{3}+2\pi r b-2\pi r^2\pm\sqrt{\left(A_c+\frac{1}{3}\right)^2-4\pi A_cr\left(\frac{1}{4}k^2t^2+\frac{1}{2}k\sqrt{h_0}t+h_0-b+r\right)}}{2\pi r}$$
Everything in this formula except $t$ and $b$, so $\pi, A_c, r, k, h_0$, are constants.
The issue I have here is that the eq. 1 has $b^2, b, \frac{db}{dt}, \left(\frac{db}{dt}\right)^2$ inside an integral and is already quite chaotic, plus the formula for $h_2$ in it is described in $b$. To make it even worse, that formula is in the form of a solution of a quadratic equation and is not in a shape which can be easily integrated.
I would like to find the approximated formula of $b\,(t)$ using Python codes.
Any help or suggestions are appreciated! Thank you!