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Question Number 201822 by MathedUp last updated on 13/Dec/23
Do Not Use sin(θ)∼θ (θ  is small Enough)  θ^  +(g/ℓ)sin(θ)=0  y′′(t)+(g/ℓ) sin(y(t))=0  y′′(t)y′(t)+(g/ℓ)sin(y(t))y′(t)=0  y′(t)y′′(t)=(1/2)∙((d  )/dt)(y′(t))^2   (g/ℓ)sin(y(t))y′(t)=−(g/ℓ)∙((d  )/dt)cos(y(t))  ∴ ((d  )/dt)[(1/2)(y′(t))^2 −(g/ℓ)cos(y(t))]=0  ∴(1/2)(y′(t))^2 −(g/ℓ)cos(y(t))=c_1   Const  y′′(t)+(g/ℓ)sin(y(t))=0→(y′(t))^2 −((2g)/ℓ)cos(y(t))=c_1   and... I can′t Sove Diff  Equa..
$$\mathrm{Do}\:\mathrm{Not}\:\mathrm{Use}\:\mathrm{sin}\left(\theta\right)\sim\theta\:\left(\theta\:\:\mathrm{is}\:\mathrm{small}\:\mathrm{Enough}\right) \\ $$$$\ddot {\theta}+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left(\theta\right)=\mathrm{0} \\ $$$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\:\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0} \\ $$$${y}''\left({t}\right){y}'\left({t}\right)+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=\mathrm{0} \\ $$$${y}'\left({t}\right){y}''\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} \\ $$$$\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=−\frac{\mathrm{g}}{\ell}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\mathrm{cos}\left({y}\left({t}\right)\right) \\ $$$$\therefore\:\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left[\frac{\mathrm{1}}{\mathrm{2}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} −\frac{\mathrm{g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)\right]=\mathrm{0} \\ $$$$\therefore\frac{\mathrm{1}}{\mathrm{2}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} −\frac{\mathrm{g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)={c}_{\mathrm{1}} \:\:\boldsymbol{\mathrm{Const}} \\ $$$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0}\rightarrow\left({y}'\left({t}\right)\right)^{\mathrm{2}} −\frac{\mathrm{2g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)={c}_{\mathrm{1}} \\ $$$$\mathrm{and}…\:\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{Sove}\:\mathrm{Diff}\:\:\mathrm{Equa}.. \\ $$
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
not only you, but all people in the  world can not solve it using elementary  functions. to find the period  you   can apply elliptic integrals.
$${not}\:{only}\:{you},\:{but}\:{all}\:{people}\:{in}\:{the} \\ $$$${world}\:{can}\:{not}\:{solve}\:{it}\:{using}\:{elementary} \\ $$$${functions}.\:{to}\:{find}\:{the}\:{period}\:\:{you}\: \\ $$$${can}\:{apply}\:{elliptic}\:{integrals}. \\ $$
Commented by MathedUp last updated on 14/Dec/23
  Thank you a lot
$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{a}\:\mathrm{lot} \\ $$

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