How to solve the differential equation of the form $a\ddot{x}-f(t)\dot{x}^{-1}+b \dot{x}^2+c=0$? Trying to obtain the equations of motion of a vehicle in 1 dimension and using Newton's 2nd law, I got the following differential equation which I was wondering if someone could help me solve it or at least approach it if not solvable analitically.
$$m\ddot{x}-\frac{P(t)}{\dot{x}}+\kappa \dot{x}^2+c=0.$$
Another alternative would be expressing P(t), the power of the motor of the car, as torque times angular velocity, $\tau \omega= \tau\ \dot{x}/r$; thus getting this:
$$m\ddot{x}-\frac{\tau(t)}{r}+\kappa \dot{x}^2+c=0.$$
At last, I'll add the initial conditions:
$x_o=0;\ \dot{x}_o=0;\ \ddot{x}_o=ct.$
The problem is that $\dot{x}_o$ can't be $0$ in either of those two diff. eqs., so just consider something similar like $0.001$.
 A: Let $x(t)=k^{-1}\ln y(t)$, $k=\frac{\kappa}{m}$ and $f(t)=\frac{\tau(t)}{mrk}-\frac{c}{mk}$. Your second equation may be written
$$\tag{1}
y''-fy=0
$$
This linear second order equation has no closed form solution for arbitrary $f(t)$. The initial conditions translate to $y(0)=1$, $y'(0)=0$. The problem is now set up for a perturbation theory treatment: I will outline the two standard methods, see eg. Bender and Orszag: Mathematical methods for full details.
Perturbation series
We attempt to find a series representation $y(t)=\sum\limits_{n=0}^\infty \epsilon^ny_n(t)$. Insert an $\epsilon$ into (1)
$$\tag{2}
y''-\epsilon fy=0
$$
Substitute the series into (2) and calculate the $y_n$ recursively. The general equations are
$$\tag{3}
y_0''=0 \qquad y_0(0)=1 \qquad y_0'(0)=0\\
y''_n=f(t)y_{n-1}(t) \qquad y_n(0)=0 \qquad y'_n(0)=0 \qquad n> 0
$$
For specific forms of $f(t)$ you might find a nice pattern. You then calculate as many $y_n$ as you can and hope that the sum $\sum\limits_{n=0}^N y_n(t)$ gives a good approximation to the solution (or you try to do something fancy with the sum).
WKB
Insert an $\epsilon^2$ into (1)
$$\tag{4}
\epsilon^2y''-fy=0
$$
Make the substitution $y(t)=\exp\left[\frac{1}{\epsilon}\sum\limits_{n=0}^\infty \epsilon^nS_n(t) \right]$ and solve for the $S_n$ recursively. The first two are well known and often give excellent approximations
$$\tag{5}
S_0(t)=\pm\int\limits^t dt' \ \sqrt{f(t')} \\
S_1(t)=-\frac{1}{4}\ln f(t)
$$
These furnish approximate solutions for all $t$ as $\epsilon \to 0$ so long as $f(t)\neq 0$. If there are points where $f(t)=0$, which seems likely given the form of $f$ with $\tau$ a function increasing from zero, then the problem becomes more interesting and you need to consider what are called turning points.
