Question

Answer Problems 1-12without referring back to the text. Fill in the blanks or answer true or false.

The first-order DE,$\frac{\mathbf{d}\mathbf{r}}{\mathbf{d}\mathbf{\theta }}\mathbf{=}\mathit{r}\mathit{\theta }\mathbf{+}\mathit{r}\mathbf{+}\mathit{\theta }\mathbf{+}\mathbf{1}$is not separable.

Short answer

False

Step-by-step solution

Definition

A first-order differential equation of the form $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathit{g}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$is said to be separable or to have separable variables.

Simplify

The given equation can be written as:

$$\begin{gathered} \frac{\mathrm{dr}}{\mathrm{d\theta }}=r\theta +r+\theta +1 \\ \frac{\mathrm{dr}}{\mathrm{d\theta }}=r\left(\theta +1\right)+\left(\theta +1\right) \\ \frac{\mathrm{dr}}{\mathrm{d\theta }}=\left(\theta +1\right)\left(\theta +1\right) \end{gathered}$$

Clearly the above equation is separable as we can separate each variable easily.

So, the given statement is false.