Exact equation

If \(F(x,y)\) is the potential, \(dF=F_x\,dx+F_y\,dy\), so exactness means \(F_x=M\) and \(F_y=N\). The equation then says \(dF=0\) and its implicit solutions are the level curves \(F(x,y)=C\). For example, \(y\,dx+x\,dy=0\) is \(d(xy)=0\), with solutions \(xy=C\).

To test exactness it is enough to compare the cross derivatives: if \(F\) exists with continuous derivatives, \(F_{xy}=F_{yx}\), that is \(M_y=N_x\). Conversely, on a rectangle, if \(M_y=N_x\) then \(F\) exists. So \((2xy+3)\,dx+(x^2-1)\,dy=0\) is exact because both derivatives equal \(2x\).

To find \(F\), integrate \(M\) with respect to \(x\), giving \(F=\int M\,dx+h(y)\), and determine \(h\) by imposing \(F_y=N\). In the example, \(F=x^2y+3x+h(y)\) and \(x^2+h'(y)=x^2-1\) gives \(h=-y\); the solution is \(x^2y+3x-y=C\). Terms that already appear are not repeated.

Typical mistakes: requiring \(M=N\) instead of \(M_y=N_x\), forgetting the function \(h(y)\) when integrating, or believing that a non-exact equation has no solution. Sometimes a factor fixes it: \(y\,dx+2x\,dy=0\) is not exact, but multiplied by \(y\) it is.