What is general solution of ode cos(y)y’=x/(x^2+1)sin(y) ?

Question : What is general solution of ode cos(y)y’=x/(x^2+1)sin(y) ?

Answer :

Given ordinary differential equation
\[cos\left(y\right)y^{‘}=\frac{x}{x^2+1}sin(y)\]We can write this as
\[\frac{d\left(\sin(y)\right)}{dx}=\frac{x}{x^2+1}\sin{\left(y\right)}\]Dividing by sin(y)
\[\frac{1}{\sin{\left(y\right)}}\frac{d\left(\sin(y)\right)}{dx}=\frac{x}{x^2+1}\]\[=>\frac{1}{\sin{\left(y\right)}}d\left(\sin{\left(y\right)}\right)=\frac{x}{x^2+1}dx\]Integrating
\[\log(\sin{\left(y\right)})=\log{\left(c\sqrt{x^2+1}\right)}\]where c is an arbitrary constant.
\[=>\sin{\left(y\right)}=c\sqrt{x^2+1}\]\[=>\ y(x)=\sin^{-1}{\left(c\sqrt{x^2+1}\right)}\]This is the required general solution of the given ode.