In this form the equation can be expressed in such a way that the coefficient of \(dx\) is only a function of \(x\) and that of \(dy\) is only a function of \(y\). The general form of such an equation is \[f\left( x \right)dx+g\left( y \right)dy=0\]we say that the variables have been separated. On integrating this reduced form we get \(\int{f(x)dx+\int{g(y)dy=c}}\), where \(c\) is any arbitrary constants. This is the general solution of the given differential equation.

Example: Solve the equation \(y(1+x)dx+x(1+y)dy=0\)

Solution: The equation can be written in variable separable form,

\[\left( \frac{1+x}{x} \right)dx+\left( \frac{1+y}{y} \right)dy=0\]

Integrating both the sides,

\[\int{\left( \frac{1+x}{x} \right)dx+}\int{\left( \frac{1+y}{y} \right)dy}=c\]

\[\Rightarrow \log x+x+\log y+y=c\]

\[\Rightarrow x+y+\log (xy)=c\]

This is the solution of the given differential equation with arbitrary constant \(c\).