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.

$\theta \mathbf{dr}\mathbf{+}\mathbf{(}\mathbf{3}\mathbf{r}\mathbf{-}\theta \mathbf{-}\mathbf{1}\mathbf{)}\mathbf{d}\theta \mathbf{=}\mathbf{0}$

Short answer

The equation is linear.

Step-by-step solution

Find if the equation is separable

Here

$$\begin{gathered} \theta \mathbf{dr}\mathbf{+}\mathbf{(}\mathbf{3}\mathbf{r}\mathbf{-}\theta \mathbf{-}\mathbf{1}\mathbf{)}\mathbf{d}\theta \mathbf{=}\mathbf{0} \\ \frac{\mathbf{d}\theta }{\mathbf{dr}}\mathbf{=}\frac{\theta }{\mathbf{1}\mathbf{+}\theta \mathbf{-}\mathbf{3}\mathbf{r}} \end{gathered}$$

This equation can be written in the form as g () h(r). so, this equation is not separable.

Determine if the equation is linear

The equation can be written as

$\frac{\mathbf{dr}}{\mathbf{d}\theta }\mathbf{+}\frac{\mathbf{3}\mathbf{r}}{\theta }\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\theta }$

Now,$\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{3}}{\theta }\mathbf{,}\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\theta }$

This equation is a linear.

Evaluate whether the equation is exact

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

$$\begin{gathered} \mathbf{M}\mathbf{(}\mathbf{r}\mathbf{,}\theta \mathbf{)}\mathbf{=}\theta \\ \mathbf{N}\mathbf{(}\mathbf{r}\mathbf{,}\theta \mathbf{)}\mathbf{=}\mathbf{3}\mathbf{r}\mathbf{-}\theta \mathbf{-}\mathbf{1} \\ \frac{\mathbf{\partial }\mathbf{M}}{\mathbf{\partial }\mathbf{\theta }}\mathbf{=}\mathbf{1} \\ \frac{\mathbf{\partial }\mathbf{N}}{\mathbf{\partial }\mathbf{r}}\mathbf{=}\mathbf{3} \\ \frac{\mathbf{\partial }\mathbf{M}}{\mathbf{\partial }\mathbf{y}}\mathbf{\ne }\frac{\mathbf{\partial }\mathbf{N}}{\mathbf{\partial }\mathbf{x}} \end{gathered}$$

This equation is not exact.

Hence, the equation is linear.