Question

In Problems 9–20, determine whether the equation is exact.

If it is, then solve it.

$\left(\frac{\mathbf{1}}{\mathbf{y}}\right)\mathbf{dx}\mathbf{-}\left(\mathbf{3}\mathbf{y}\mathbf{-}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right)\mathbf{dy}\mathbf{=}\mathbf{0}$

Short answer

The equation is not exact.

Step-by-step solution

Evaluate whether the equation is exact

Here$\left(\frac{\mathbf{1}}{\mathbf{y}}\right)\mathbf{dx}\mathbf{-}\left(\mathbf{3}\mathbf{y}\mathbf{-}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right)\mathbf{dy}\mathbf{=}\mathbf{0}$

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

$$\begin{gathered} \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left(\frac{\mathbf{1}}{\mathbf{y}}\right) \\ \mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left(\mathbf{3}\mathbf{y}\mathbf{-}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right) \\ \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{y}}^{\mathbf{2}}} \\ \frac{\partial \mathbf{N}}{\partial \mathbf{x}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{y}}^{\mathbf{2}}} \\ \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\ne \frac{\partial \mathbf{N}}{\partial \mathbf{x}} \end{gathered}$$

Hence the equation is not exact.