Question Number 154785 by SANOGO last updated on 21/Sep/21
$${soit}:\:{y}'+{tan}\left({x}\right){y}={sin}\left(\mathrm{2}{x}\right)\:,{avec} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{1}\:\:\:{alors}\:{f}\left(\pi\right)=? \\ $$
Commented by SANOGO last updated on 21/Sep/21
$${merci}\:{bien}\:{le}\:{dur} \\ $$
Commented by tabata last updated on 21/Sep/21
$$\boldsymbol{{you}}\:\boldsymbol{{are}}\:\boldsymbol{{welcome}} \\ $$
Commented by tabata last updated on 21/Sep/21
$$\boldsymbol{{p}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)\:\:\:\:,\:\:\:\boldsymbol{{Q}}\left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{sin}}\left(\mathrm{2}\boldsymbol{{x}}\right) \\ $$$$ \\ $$$$\left(\boldsymbol{{I}}.\boldsymbol{{f}}\right)\:=\:\boldsymbol{{e}}^{\int\:\boldsymbol{{p}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}} \:=\:\boldsymbol{{e}}^{\int\:\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)} \:=\:\boldsymbol{{e}}^{−\boldsymbol{{ln}}\mid\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\mid} \:=\:\boldsymbol{{sec}}\left(\boldsymbol{{x}}\right) \\ $$$$ \\ $$$$\boldsymbol{{y}}\:=\:\frac{\int\:\left(\boldsymbol{{I}}.\boldsymbol{{f}}\right)\:\boldsymbol{{Q}}\:\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}}{\left(\boldsymbol{{I}}.\boldsymbol{{f}}\right)}\:=\:\frac{\int\:\mathrm{2}\:\boldsymbol{{sin}}\:\left(\boldsymbol{{x}}\right)\:\boldsymbol{{dx}}\:}{\boldsymbol{{sec}}\left(\boldsymbol{{x}}\right)}\:=\:−\:\mathrm{2}\:\boldsymbol{{cos}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)\:+\:\boldsymbol{{c}}\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right) \\ $$$$ \\ $$$$\therefore\:\:\boldsymbol{{y}}\:=\:−\:\mathrm{2}\:\boldsymbol{{cos}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)\:+\:\boldsymbol{{c}}\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right) \\ $$$$ \\ $$$$\boldsymbol{{f}}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:\Rightarrow\:\mathrm{1}\:=\:−\mathrm{2}\:+\:\boldsymbol{{c}}\:\Rightarrow\:\boldsymbol{{c}}\:=\:\mathrm{3} \\ $$$$ \\ $$$$\therefore\:\boldsymbol{{y}}\:=\:−\:\mathrm{2}\:\boldsymbol{{cos}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)\:+\:\mathrm{3}\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right) \\ $$$$ \\ $$$$\boldsymbol{{f}}\:\left(\:\boldsymbol{\pi}\:\right)\:=\:−\:\mathrm{2}\:\left(\:\boldsymbol{{cos}}\left(\boldsymbol{\pi}\right)\right)^{\mathrm{2}} \:+\:\mathrm{3}\:\boldsymbol{{cos}}\left(\boldsymbol{\pi}\right) \\ $$$$ \\ $$$$\boldsymbol{{f}}\:\left(\:\boldsymbol{\pi}\:\right)\:=\:−\:\mathrm{2}\:−\:\mathrm{3}\:=\:−\mathrm{5}\: \\ $$$$ \\ $$$$\langle\:\boldsymbol{{M}}\:.\:\boldsymbol{{T}}\:\:\rangle \\ $$