Question

find an integrating factor and solve the given equation
\[1+(\frac{x}{y}-siny)y'=0\]
Please, help

Answer 1

\[1+\left( \frac{ x }{ y }-\sin y \right)y \prime=0\]
\[\left( \frac{ x }{ y }-\sin y \right)\frac{ dy }{ dx }=-1\]
\[\frac{ x }{ y }-\sin y=-\frac{ dx }{dy }\]
\[\frac{ dx }{ dy }+\frac{ 1 }{ y }x=\sin y\]
\[I.F.=e ^{\int\limits \frac{ 1 }{ y }dy}=e ^{\ln y}=y\] (assuming y>0)
\[C.S. is x y=\int\limits y \sin y dy +c\]
\[xy=y (-\cos y)-\int\limits 1*\left(- \cos y \right)dy+c\]
\[xy=-y \cos y+\sin y+c\]

Answer 2

Thanks for reply and sorry for not being here to get help. The net was so bad here. I saw you wrote something but couldn't access to the net. Again, thank you very much. I appreciate.

Answer 3

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