Question

Q\Determine the following ODEs are exact or not? Why?
(x/2)(cosy)y´=-1

Answer 1

Given $\frac{x}{2}\cos(y)y'=-1$ which can be expressed as

$\frac{x}{2}\cos(y)\frac{dy}{dx}=-1$

Equivalently we have

$\frac{x}{2}\cos(y)dy + dx=0$

The ODE is not exact since $\frac{\partial(\frac{x}{2}\cos (y))}{\partial x}=\frac{1}{2}\cos(y) \ne \frac{\partial(1)}{\partial y}=0$