Chapter 2.4, Problem 42E

To determine

(a)

To find: The function $M(x,y)$ so that differential equation $M(x,y)dx+\left(x{e}^{xy}+2xy+\frac{1}{x}\right)dy=0$ is exact.

To determine

(b)

To find: The function $N\left(x,y\right)$ so that differential equation $\left(x^{-1/2}{y}^{1/2}+\frac{x}{x^2+y}\right)dx+N\left(x,y\right)dy=0$ is exact.