Esmeralda Lane

2022-07-06

How to approach the ODE ${y}^{\prime }+\mathrm{sin}\left(x+y\right)=\mathrm{sin}\left(x-y\right)$

esperoanow

Expert

${y}^{\prime }+\mathrm{sin}\left(x+y\right)=\mathrm{sin}\left(x-y\right)$
${y}^{\prime }=\mathrm{sin}\left(x-y\right)-\mathrm{sin}\left(x+y\right)=-2\mathrm{cos}x\mathrm{sin}y$
$\frac{dy}{dx}=-2\mathrm{cos}x\mathrm{sin}y$
$\frac{dy}{\mathrm{sin}y}=-2\mathrm{cos}x\phantom{\rule{thinmathspace}{0ex}}dx$
$\mathrm{csc}y\phantom{\rule{thinmathspace}{0ex}}dy=-2\mathrm{cos}x\phantom{\rule{thinmathspace}{0ex}}dx$
Integrating both sides, we get
$\mathrm{ln}|\mathrm{csc}y-\mathrm{cot}y|=-2\mathrm{sin}x+c$
where c is a constant of integration.

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