Question

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

The linear DE${\mathit{a}}_{\mathbf{1}}\left(x\right){\mathit{y}}^{\cancel{\mathbf{c}}}\mathbf{+}{\mathit{a}}_{\mathbf{0}}\left(x\right)\mathit{y}\mathbf{=}\mathbf{0}$ is also separable.

Short answer

True.

Step-by-step solution

Definition

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

Solve given equation

The given linear DE can be written as:

$\begin{array}{rcl}{a}_1\left(x\right)y\text{'}+a_0\left(x\right)y& =& 0\\ a_1\left(x\right)y\text{'}& =& -a_0\left(x\right)y\\ y\text{'}& =& -\frac{a_0\left(x\right)}{a_1\left(x\right)}y\\ \frac{dy}{dx}& =& -\frac{a_0\left(x\right)}{a_1\left(x\right)}y\\ \frac{dy}{y}& =& -\frac{a_0\left(x\right)}{a_1\left(x\right)}dx\end{array}$

Which is separable so given statement is true.