91Ó°ÊÓ

Let $\mathrm{y}\left(\mathrm{t}\right)$be a solution to the differential equation $\mathrm{y}"=\mathrm{f}\left(\mathrm{y}\right)$, where f(y) is a continuous function that does not depend on y’ or the independent variable t.

Let F(y) is an indefinite integral of $\mathrm{f}\left(\mathrm{y}\right)$i.e. $\mathrm{f}\left(\mathrm{y}\right)=\frac{\mathrm{d}}{\mathrm{dy}}\mathrm{F}\left(\mathrm{y}\right)$. Then the quantity$\mathrm{E}\left(\mathrm{t}\right)=\frac{1}{2}\mathrm{y}\text{'}{\left(\mathrm{t}\right)}^2-\mathrm{F}\left(\mathrm{y}\left(\mathrm{t}\right)\right)$ is constant; i.e.role="math" localid="1655365549886" $\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{E}\left(\mathrm{t}\right)=0$ .

Referring from Problem 8: $\frac{1}{2}\mathrm{θ}\text{'}{\left(\mathrm{t}\right)}^2-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õθ}=\mathrm{constant}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}…\left(1\right)$

To prove: the pendulum is released from rest at the angle $0<\mathrm{Î\pm }<\mathrm{Ï€}$, then$\left|\mathrm{θ}\left(\mathrm{t}\right)\right|≤\mathrm{Î\pm }$for all t.

The initial conditions are $\mathrm{θ}\left(0\right)=\mathrm{Î\pm },\mathrm{θ}\text{'}\left(0\right)=0$.

Let us take role="math" localid="1655365792527" $\mathrm{constant}=\mathrm{k}$. Then,

role="math" localid="1655365833327" $\frac{1}{2}\mathrm{θ}\text{'}{\left(\mathrm{t}\right)}^2-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õθ}=\mathrm{k}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â€‰â€‰â€‰â€‰}…\left(2\right)$

Now, implement the initial conditions here.

role="math" localid="1655365997812" $\begin{array}{c}\frac{1}{2}\mathrm{θ}\text{'}{\left(\mathrm{t}\right)}^2-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õθ}=\mathrm{k}\\ \frac{1}{2}\mathrm{θ}\text{'}{\left(0\right)}^2-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õθ}\left(0\right)=\mathrm{k}\\ 0-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õÎ\pm }=\mathrm{k}\\ \mathrm{k}=-\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õÎ\pm }\end{array}$

Now, substitute the value of k in equation (2).

Here ${(\mathrm{θ}\text{'})}^2$can’t be a negative number. So, we can write this condition as:

$\begin{array}{c}\frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õθ}â‰\yen \frac{\mathrm{g}}{\mathrm{â„“}}\mathrm{³¦´Ç²õÎ\pm }\\ \mathrm{³¦´Ç²õθ}â‰\yen \mathrm{³¦´Ç²õÎ\pm }\end{array}$

Multiply${\cos }^{−1}$ on both sides.

Hence it is proved that $\left|\mathrm{θ}\left(\mathrm{t}\right)\right|≤\mathrm{Î\pm }$.