What is the soln to the differential equation : $x'''+x''+xx' = 0$? $$x'''+x''+xx' = 0, \quad x' = \frac{\mathrm{d}x}{\mathrm{d}t}.$$
Is there an analytic soln for this?
I am very new to numerical methods also, how do I solve this in python, if a numerical method is the only possibility?
 A: $$\frac{d^3x}{dt^3}+\frac{d^2x}{dt^2}+x\frac{dx}{dt}=0 \tag 1$$
$$\frac{d^2x}{dt^2}+\frac{dx}{dt}+\frac{x^2}{2}=c_1$$
This is an ODE of autonomous kind. The usual way to solve it is the change of function :
$$\frac{dx}{dt}=y(x)\quad;\quad \frac{d^2x}{dt^2}=\frac{dy}{dx}\frac{dx}{dt}=y\frac{dy}{dx}$$
$$y\frac{dy}{dx}+y+\frac{x^2}{2}=c_1\tag 2$$
This is an Abel's differential equation. Most of them are not analytically solvable in form of a finite number of standard functions. See https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf
I didn't check if this Abel's ODE belongs or not to the solvables ones. Even if the answer was yes, this would be much more complicated than numerical solving.
In practice using a numerical method of solving is recommended. Especially in the present case with a first order non-linear ODE $(2)$ which is simpler than the original third order non-linear ODE $(1)$. 
A: So, given the ordinary differential equation:
$$\dddot{x}(t) = - \ddot{x}(t) - x(t) \cdot \dot{x}(t)\,, $$
we can solve it numerically by applying, for example, the method of Runge-Kutta-4.
To do this, first of all, we must assume a reasonable number of initial conditions, for example:
$$x(0) = 1\,, \; \; \; \; \; \; \dot{x}(0) = 3\,, \; \; \; \; \; \; \ddot{x}(0) = 2\,,$$
then write this third-order differential equation in an equivalent system of three differential equations of the first order, naturally with the respective initial conditions:
$$
\begin{cases}
\dot{x}(t) = y(t) \\
\dot{y}(t) = z(t) \\
\dot{z}(t) = - z(t) - x(t) \cdot y(t) \\
x(0) = 1 \\
y(0) = 3 \\
z(0) = 2 
\end{cases} \;.
$$
At this point it's necessary to choose the spreadsheet where to implement the aforementioned numerical method, I trivially choose Microsoft Excel. In particular, I set the sheet as follows:

where I highlighted in light red the initial conditions and in light green the step with which to discretize the integration interval, assumed in the following equal to $0 \le t \le 2.50\,s$.
Now it's time to fill in the first row of the table as follows:

therefore it is time to compile also the second line, starting from the first block:

We're done writing. All that remains is to complete the compilation of the second line by dragging down the first, then highlight the second complete line by dragging until $t = 2.50\,s$, obtaining:

Dulcis in fundo, it's sufficient to place on the abscissas $t$ and on the ordinates $x(t)$, obtaining:

All this to show that it isn't necessary to have a sophisticated software to solve numerically an ordinary differential equation, even if, for example, writing in Wolfram Mathematica:
sol = NDSolve[{x'''[t] == -x''[t] - x[t] x'[t], 
               x[0] == 1, x'[0] == 3, x''[0] == 2}, 
               x[t], {t, 0, 2.5}];

Plot[Evaluate[x[t] /. sol], 
     {t, 0, 2.5}, 
     AxesOrigin -> {0, 0}]

you get immediately:

but perhaps, especially at the beginning, it is better to get your hands dirty in Excel to put the theory into practice, which must be studied in depth!
