Question

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

Every autonomous$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$is separable.

Short answer

True

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 differential equation is $\frac{\mathrm{dy}}{\mathrm{dx}}=f\left(\mathrm{y}\right)$

It can be written as:

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}=f\left(\mathrm{y}\right) \\ \frac{\mathrm{dy}}{f\left(\mathrm{y}\right)}=dx \end{gathered}$$

So, the given statement is true.