# Solve differential equation y'+y\cot(x)=sin(2x)

Solve differential equation${y}^{\prime }+y\mathrm{cot}\left(x\right)=\mathrm{sin}\left(2x\right)$
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Arham Warner
${y}^{\prime }+P\left(x\right)y=Q\left(x\right)$
$I.F.={e}^{\int Pdx}$
$y\left(I.F.\right)=\int \left(I.F.\right)Q\left(x\right)dx+c$

$I.F.={e}^{\int \mathrm{cot}\left(x\right)dx}={e}^{\mathrm{log}\left(\mathrm{sin}\left(x\right)\right)}=\mathrm{sin}\left(x\right)$
$y\left(\mathrm{sin}\left(x\right)\right)=\int \mathrm{sin}\left(x\right)\mathrm{sin}\left(2x\right)dx+c$
$y\left(\mathrm{sin}\left(x\right)\right)=\int \mathrm{sin}\left(x\right)2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)dx+c$
$y\left(\mathrm{sin}\left(x\right)\right)=2\int {\mathrm{sin}}^{2}\left(x\right)\mathrm{cos}\left(x\right)dx+c$
Fisrt solve the integral
Let
$\mathrm{sin}\left(x\right)=t⇒\mathrm{cos}\left(x\right)dx=dt$
$\int {\mathrm{sin}}^{2}\left(x\right)\mathrm{cos}\left(x\right)dx=\int {t}^{2}dt=\frac{{t}^{3}}{3}+{c}_{1}=\frac{{\mathrm{sin}}^{3}\left(x\right)}{3}+{c}_{1}$
$y\left(\mathrm{sin}\left(x\right)\right)=2\frac{{\mathrm{sin}}^{3}\left(x\right)}{3}+{c}_{1}+c$