Question #293048

 4. Find the general solution of the differential equation dy/dx + y cot x = 1, recalling that cot x = cos x / sin x 


1
Expert's answer
2022-02-03T16:29:16-0500

We have;

dydx+ycot(x)=1dydx+ycos(x)sin(x)=1\displaystyle \frac{dy}{dx}+y\cot (x)=1\Rightarrow \frac{dy}{dx}+y\frac{\cos(x)}{\sin(x)}=1

The Integrating Factor (I.F) of the DE is;

ecos(x)sin(x) dx=elnsin(x)=sin(x)\displaystyle e^{\int\frac{\cos (x)}{\sin(x)}\ dx}=e^{\ln|\sin(x)|}=\sin(x)

Multiplying the I.F by the given DE yields;

(dydx+ycos(x)sin(x))sin(x)=1×sin(x)ddx(ysin(x))=sin(x)ysin(x)=sin(x) dxysin(x)=cos(x)+a, where a is an arbitrary constant.y=acos(x)sin(x)\displaystyle \left(\frac{dy}{dx}+y\frac{\cos(x)}{\sin(x)}\right)\sin(x)=1\times\sin(x)\\ \Rightarrow \frac{d}{dx}(y\sin(x))=\sin(x)\\ \Rightarrow y\sin(x)=\int\sin(x)\ dx\\ \Rightarrow y\sin(x)=-\cos(x)+a,\text{ where a is an arbitrary constant.}\\ \Rightarrow y=\frac{a-\cos(x)}{\sin(x)}


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