Question Number 201740 by MathedUp last updated on 12/Dec/23
![y′′(t)+(g/ℓ)∙sin(y(t))=0 L_s {y′′(t)}+(g/ℓ)L_s {sin(y(t))}=0 ∫_0 ^( ∞) y′′(t)e^(−st) dt+(g/ℓ)∫_0 ^( ∞) sin(y(t))e^(−st) dt=0 ∫_0 ^( ∞) {y′′(t)+(g/ℓ)[y(t)−(((y(t))^3 )/(3!))+(((y(t))^5 )/(5!))∙∙∙∙]}e^(−st) dt=0 ∫_0 ^( ∞) ′′=K(s)=0 Can′t Solve this equation Should I change sin(y(t))=((e^(−iy(t)) −e^(iy(t)) )/2)i and e^(iy(t)) =u(t) ....??](https://www.tinkutara.com/question/Q201740.png)
$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\centerdot\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0} \\ $$$$\boldsymbol{\mathcal{L}}_{{s}} \left\{{y}''\left({t}\right)\right\}+\frac{\mathrm{g}}{\ell}\boldsymbol{\mathcal{L}}_{{s}} \left\{\mathrm{sin}\left({y}\left({t}\right)\right)\right\}=\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {y}''\left({t}\right){e}^{−{st}} \mathrm{d}{t}+\frac{\mathrm{g}}{\ell}\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\left({y}\left({t}\right)\right){e}^{−{st}} \mathrm{d}{t}=\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\left\{{y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\left[{y}\left({t}\right)−\frac{\left({y}\left({t}\right)\right)^{\mathrm{3}} }{\mathrm{3}!}+\frac{\left({y}\left({t}\right)\right)^{\mathrm{5}} }{\mathrm{5}!}\centerdot\centerdot\centerdot\centerdot\right]\right\}{e}^{−{st}} \mathrm{d}{t}=\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:''={K}\left({s}\right)=\mathrm{0} \\ $$$$\mathrm{Can}'\mathrm{t}\:\mathrm{Solve}\:\mathrm{this}\:\mathrm{equation} \\ $$$$\mathrm{Should}\:\mathrm{I}\:\mathrm{change}\:\mathrm{sin}\left({y}\left({t}\right)\right)=\frac{{e}^{−\boldsymbol{{i}}{y}\left({t}\right)} −{e}^{\boldsymbol{{i}}{y}\left({t}\right)} }{\mathrm{2}}\boldsymbol{{i}} \\ $$$$\mathrm{and}\:{e}^{\boldsymbol{{i}}{y}\left({t}\right)} ={u}\left({t}\right)\:….??\: \\ $$