Answer to Question #293048 in Differential Equations for jacob

Question

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

Answer 1

We have;

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

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

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

Multiplying the I.F by the given DE yields;

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