Editing Autonomous system (mathematics) (section)

====Special case: {{math|1=''x''&Prime; = ''x''&prime;<sup>''n''</sup> ''f''(''x'')}}====

Using the above approach, the technique can extend to the more general equation

<math display="block">\frac{d^2 x}{d t^2} = \left(\frac{d x}{d t}\right)^n f(x)</math>

where <math>n</math> is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of <math>x'</math>. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:

<math display="block">\begin{align}
&- \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \left(\frac{d t}{d x}\right)^{-n} f(x) \\[4pt]
&- \left(\frac{d t}{d x}\right)^{n - 3} \frac{d^2 t}{d x^2} = f(x) \\[4pt]
&\frac{d}{d x}\left(\frac{1}{2 - n}\left(\frac{d t}{d x}\right)^{n - 2}\right) = f(x) \\[4pt]
&\left(\frac{d t}{d x}\right)^{n - 2} = (2 - n) \int f(x) dx + C_1 \\[2pt]
&t + C_2 = \int \left((2 - n) \int f(x) dx + C_1\right)^{\frac{1}{n - 2}} dx
\end{align}</math>

The right will carry +/− if <math>n</math> is even. The treatment must be different if <math>n = 2</math>:

<math display="block">\begin{align}
- \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2}    &= f(x) \\
-\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) &= f(x) \\
\frac{d t}{d x} &= C_1 e^{-\int f(x) dx} \\
t + C_2 &= C_1 \int e^{-\int f(x) dx} dx
\end{align}</math>