[Math] Solve $\theta”+g\sin(\theta)=0$

ordinary differential equations

I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum:

$\frac{\mathrm d^2 \theta}{\mathrm dt^2}+g\sin\theta=0$

How can I solve the above equation?

Best Answer

Use substitution : $\theta' =v$ ,therefore we have that :

$$\theta''=\frac{dv}{dt}\cdot \frac{dt}{d\theta}\cdot \frac{d\theta}{dt} \Rightarrow \theta''=\frac{dv}{d\theta}\cdot v \Rightarrow \theta''=v'\cdot v$$

where $v$ is function in terms of variable $\theta$ .So differential equation becomes :

$v' \cdot v +g \cdot \sin \theta=0$

which is separable differential equation .

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