Question 1167385
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y'\ =\ \csc(x)\ -\ y\,\cot(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{dy(x)}{dx}\ +\ y\,\cot(x)\ =\ \csc(x)]


Let *[tex \Large \mu\ =\ e^{\int\,\cot(x)dx}\ =\ \sin(x)] and multiply through


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \sin(x)\frac{dy(x)}{dx}\ +\ \cos(x)y(x)\ =\ 1]


Substitute *[tex \Large \frac{d\sin(x)}{dx}\ =\ \cos(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \sin(x)\frac{dy(x)}{dx}\ +\ \frac{d\sin(x)}{dx}y(x)\ =\ 1]


Reverse Product Rule


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\[\sin(x)y(x)\]\ =\ 1]


Integrate wrt *[tex \Large x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \int\,\frac{d}{dx}\[\sin(x)y(x)\]\,dx\ =\ \int\,1\,dx ]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)y(x)\ =\ x\ +\ C]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y(x)\ =\ \csc(x)(x\ +\ C)]


																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
*[illustration darwinfish]


I > Ø
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
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