Nonlinear system: Difference between revisions - Wikipedia

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where gravity points "downwards" and <math>\theta</math> is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation is to use <math>d\theta/dt</math> as an [[integrating factor]], which would eventually yield

:<math>\int {\frac{d \theta}{\sqrt{C_0 + 2 \cos(\theta)}}} = t + C_1\,</math>

which is an implicit solution involving an [[elliptic integral]]. This "solution" generally does not have many uses because most of the nature of the solution is hidden in the [[nonelementary integral]] (nonelementary even if <math>C_0 = 0</math>).