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Question Number 201822 by MathedUp last updated on 13/Dec/23

$$\mathrm{Do}\mathrm{Not}\mathrm{Use}\sin \left(\theta \right)\sim \theta \left(\theta \mathrm{is}\mathrm{small}\mathrm{Enough}\right)$$$$\ddot{\theta }+\frac{\mathrm{g}}{\ell }\sin \left(\theta \right)=0$$$$y^{″}\left(t\right)+\frac{\mathrm{g}}{\ell }\sin \left(y\left(t\right)\right)=0$$$$y^{″}\left(t\right)y^{\prime }\left(t\right)+\frac{\mathrm{g}}{\ell }\sin \left(y\left(t\right)\right)y^{\prime }\left(t\right)=0$$$$y^{\prime }\left(t\right)y^{″}\left(t\right)=\frac{1}{2}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{\left(y^{\prime }\left(t\right)\right)}^2$$$$\frac{\mathrm{g}}{\ell }\sin \left(y\left(t\right)\right)y^{\prime }\left(t\right)=-\frac{\mathrm{g}}{\ell }\cdot \frac{\mathrm{d}}{\mathrm{d}t}\cos \left(y\left(t\right)\right)$$$$\therefore \frac{\mathrm{d}}{\mathrm{d}t}[\frac{1}{2}{\left(y^{\prime }\left(t\right)\right)}^2-\frac{\mathrm{g}}{\ell }\cos \left(y\left(t\right)\right)]=0$$$$\therefore \frac{1}{2}{\left(y^{\prime }\left(t\right)\right)}^2-\frac{\mathrm{g}}{\ell }\cos \left(y\left(t\right)\right)=c_1\mathbf{Const}$$$$y^{″}\left(t\right)+\frac{\mathrm{g}}{\ell }\sin \left(y\left(t\right)\right)=0\to {\left(y^{\prime }\left(t\right)\right)}^2-\frac{2\mathrm{g}}{\ell }\cos \left(y\left(t\right)\right)=c_1$$$$\mathrm{and}...\mathrm{I}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{Sove}\mathrm{Diff}\mathrm{Equa}..$$

Commented by mr W last updated on 13/Dec/23

Commented by mr W last updated on 13/Dec/23

Commented by mr W last updated on 14/Dec/23

$$notonlyyou,butallpeopleinthe$$$$worldcannotsolveitusingelementary$$$$functions.tofindtheperiodyou$$$$canapplyellipticintegrals.$$

Commented by MathedUp last updated on 14/Dec/23

$$$$$$\mathrm{Thank}\mathrm{you}\mathrm{a}\mathrm{lot}$$

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