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.

$\mathbf{xydx}\mathbf{+}\mathbf{dy}\mathbf{=}\mathbf{0}$

Short answer

The equation is separable and also linear.

Step-by-step solution

Find if the equation is separable

Here

$$\begin{gathered} \mathbf{xy}\mathbf{dx}\mathbf{+}\mathbf{dy}\mathbf{=}\mathbf{0} \\ \frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\mathbf{xy} \end{gathered}$$

This equation can be written in the form as g(x) h(y).so, this equation is Separable.

Determine if the equation is linear

The equation can be written as

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{xy}\mathbf{=}\mathbf{0}$

Now,$\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{x}\mathbf{,}\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{0}$

This equation is a linear.

Evaluate whether the equation is exact

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

$$\begin{gathered} \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{xy} \\ \mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{1} \\ \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\mathbf{=}\mathbf{x} \\ \frac{\partial \mathbf{N}}{\partial \mathbf{x}}=0 \\ \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\ne \frac{\partial \mathbf{N}}{\partial \mathbf{x}} \end{gathered}$$

This equation is not exact.

Hence, the equation is separable and also linear.