I made my own air cannon at home and would like to determine its launch velocity. I worked out the following differential equation:
$$\frac{d^2}{dt^2} x(t) = \frac{A}{m} \left(\frac{P_0V_0}{V_0 + Ax(t)} - P_\text{atm}\right) $$
How would I solve this equation?
Can anyone verify whether or not this is a solution?
$$x(t) = \frac{V_0}{A}\left\{1 - \exp\left[\text{erfi}^{-1}\left( At \sqrt{\frac{2P_0}{\pi V_0m}}\right)^2 \right] \right\}$$
Getting rid of strange constants, your differential equation can be written as $$ x''=\frac{c_1}{c_2+x}+c_3. $$ Multiply by $x'$ and integrate, to get $$ \frac{1}{2}(x')^2=c_1\log(c_2+x)+c_3x+c_4 $$ Separating variables, you need to calculate an integral of the form $$ \int \frac{1}{\sqrt{2c_1\log(c_2+x)+2c_3x+2c_4}}\,dx. $$ I suggest that you use your air cannon for that purpose.