\(sinx dy/dx+ ycosx=0\)

\(sinx dy= -ycosx dx\)

\(dy/y= - cosx/sinx dx\)

\(int dy/y= -int cosx/sinx dx\)

\(ln abs(y)= -ln abs(sinx)+ln abs(C)\)

\(lny= ln (C/sinx)\)

\(ysinx= C\)

Now, We are applying the given Initial Condition is as follow

\(y((7pi)/6)= -2\)

\(-2*sin((7pi)/6)= C\) \(-2* -1= C\) \(:.sin(7pi/6)= -1\)

C=2

\(ysinx= 2\)

\(sinx dy= -ycosx dx\)

\(dy/y= - cosx/sinx dx\)

\(int dy/y= -int cosx/sinx dx\)

\(ln abs(y)= -ln abs(sinx)+ln abs(C)\)

\(lny= ln (C/sinx)\)

\(ysinx= C\)

Now, We are applying the given Initial Condition is as follow

\(y((7pi)/6)= -2\)

\(-2*sin((7pi)/6)= C\) \(-2* -1= C\) \(:.sin(7pi/6)= -1\)

C=2

\(ysinx= 2\)