Question

In problems 1-8 Classify the equation as separable, linear, exact, or none of these. Notice that some equations may have more than one classification.

$\left[\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{ycos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\right]\mathbf{dx}\mathbf{+}\left[\mathbf{xcos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{-}\mathbf{2}\mathbf{y}\right]\mathbf{dy}\mathbf{=}\mathbf{0}$

Short answer

The equation is exact.

Step-by-step solution

Find if the equation is separable

Here

$$\begin{gathered} \left[\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\right]\mathbf{dx}\mathbf{+}\left[\mathbf{x}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{-}\mathbf{2}\mathbf{y}\right]\mathbf{dy}\mathbf{=}\mathbf{0} \\ \frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}}{\mathbf{2}\mathbf{y}\mathbf{-}\mathbf{x}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}} \end{gathered}$$

This equation can be written in the form as g(x) h(y). so, this equation is not Separable

Determine if the equation is linear.

This equation is not linear of y and x because the RHS is not linear with respect to y and x respectively.

Evaluate whether the equation is exact.

The condition for exact is$\frac{\partial \mathbf{M}}{\partial \mathbf{y}}=\frac{\partial \mathbf{N}}{\partial \mathbf{x}}$ .

$$\begin{gathered} \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{cos}\mathbf{xy} \\ \mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{x}\mathbf{cos}\mathbf{xy}\mathbf{-}\mathbf{2}\mathbf{y} \\ \frac{\mathbf{\partial }\mathbf{M}}{\mathbf{\partial }\mathbf{y}}\mathbf{=}\mathbf{cos}\mathbf{}\mathbf{xy}\mathbf{-}\mathbf{xy}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{xy}\mathbf{=}\frac{\mathbf{\partial }\mathbf{N}}{\mathbf{\partial }\mathbf{x}} \end{gathered}$$

This equation is exact.

Hence, the equation is exact.