Solve the differential equation:
\left ( 1+x^{2} \right )dy+2xy\, dx= \cot x\: dx

 

 

 

 
 
 
 
 

Answers (1)

\left ( 1+x^{2} \right )dy+2xydx= \cot x\, dx\: \: \left [ given \right ]
\frac{dy}{dx}+\frac{2xy}{1+x^{2}}= \frac{\cot x}{1+x^{2}}
The linear differential equation is
IF=e^{ \int pdx}= e^{\int \frac{2x}{1+x^{2}}dx}= 1+x^{2}

The general solution is:
y\left ( 1+x^{2} \right )= \int \left [ \frac{\cot x}{1+x^{2}}\left ( 1+x^{2} \right ) \right ]dx+c
= y\left ( 1+x^{2} \right )= \log \left [ \sin x \right ]+c

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