Multiply this equation by y:

\(y dx + (x − y \sin y)dy = 0\)

Now we will try to find a function F: \(RR^{2} \rightarrow R\) such that

\(\frac{\partial F}{\partial x}=y\)

and \(\frac{\partial F}{\partial x} = x - y \sin y\) (1)

Start with \(\frac{\partial F}{\partial x} = y\)

and integrate with respect to x:

\(F(x, y) = \int y dx = xy + C(y) (2)\)

where C(y) is a constant with respect to x (an integrating constant!). From (2)

\(\frac{\partial F}{\partial x} = x + C'(y)\)

Using (1) now, we get that

\(x + C'(y) = x - y \sin y \Rightarrow C'(y) = -y \sin y\)

Thus,

\(C(y) = \int -y \sin y dy\)

\(= - \int y \sin y dy\)

\(= {(u = y \Rightarrow du = dy),(dv = \sun\ y\ dy \Rightarrow v = -\cos y)}\)

\(= -(y(-\cos\ y) - \int - \cos\ y\ dy)\)

\(= y \cos y - \int \cos y dy\)

\(= y \cos y - \sim y + D,\)

where D is a constant.

Therefore,

\(F(x, y) = xy + C(y) = xy + y \cos y - \sin y + D\)

Now the solution of the initial differential equation is given by \(F(x, y) = 0\),

so \(xy + y \cos y - \sin y = - D = C,\)

where C is some constant.

\(y dx + (x − y \sin y)dy = 0\)

Now we will try to find a function F: \(RR^{2} \rightarrow R\) such that

\(\frac{\partial F}{\partial x}=y\)

and \(\frac{\partial F}{\partial x} = x - y \sin y\) (1)

Start with \(\frac{\partial F}{\partial x} = y\)

and integrate with respect to x:

\(F(x, y) = \int y dx = xy + C(y) (2)\)

where C(y) is a constant with respect to x (an integrating constant!). From (2)

\(\frac{\partial F}{\partial x} = x + C'(y)\)

Using (1) now, we get that

\(x + C'(y) = x - y \sin y \Rightarrow C'(y) = -y \sin y\)

Thus,

\(C(y) = \int -y \sin y dy\)

\(= - \int y \sin y dy\)

\(= {(u = y \Rightarrow du = dy),(dv = \sun\ y\ dy \Rightarrow v = -\cos y)}\)

\(= -(y(-\cos\ y) - \int - \cos\ y\ dy)\)

\(= y \cos y - \int \cos y dy\)

\(= y \cos y - \sim y + D,\)

where D is a constant.

Therefore,

\(F(x, y) = xy + C(y) = xy + y \cos y - \sin y + D\)

Now the solution of the initial differential equation is given by \(F(x, y) = 0\),

so \(xy + y \cos y - \sin y = - D = C,\)

where C is some constant.