Exercise · ZW-EXACT-EQUATION-010 · ★★★ Advanced
If on a region \(\dfrac{\partial M}{\partial y}\ne\dfrac{\partial N}{\partial x}\), what follows?
Choose one option.
A The equation has no solution B There is no function \(F\) with \(F_x=M\) and \(F_y=N\); the equation is not exact C The equation is linear D The equation is exact but with another potential Show answer B. There is no function \(F\) with \(F_x=M\) and \(F_y=N\); the equation is not exact← Back to the question
Explanation If \(F\) existed, the mixed partials \(F_{xy}\) and \(F_{yx}\) would agree, and they equal \(M_y\) and \(N_x\). Not being exact does not mean there are no solutions.
An exact equation is a differential equation \(M\,dx+N\,dy=0\) whose left side is the total differential of a function \(F(x,y)\).
\[\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}\]
More practice problems: exact equation