Question

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

If it is, then solve it.

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

Short answer

The solution is ${\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\sqrt[\mathbf{3}]{\mathbf{y}}\mathbf{=}\mathbf{C}$.

Step-by-step solution

Evaluate whether the equation is exact

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

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

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

This equation is exact.

Find the value of F (x, y)

Here

$$\begin{gathered} \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{(}{\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}\mathbf{3}{\mathbf{x}}^{\mathbf{2}}\mathbf{)} \\ \mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\int \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{dx}\mathbf{+}\mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)} \\ \mathbf{=}\int \mathbf{(}{\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}\mathbf{3}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}\mathbf{dx}\mathbf{+}\mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)} \\ \mathbf{=}{\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)} \end{gathered}$$

Determine the value of g(y)

$$\begin{gathered} \frac{\partial \mathbf{F}}{\partial \mathbf{y}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \\ {\mathbf{e}}^{\mathbf{x}}\mathbf{cos}\mathbf{y}\mathbf{+}\mathbf{g}\mathbf{\text{'}}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\left({\mathbf{e}}^{\mathbf{x}}\mathbf{cos}\mathbf{y}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{-}\frac{\mathbf{2}}{\mathbf{3}}}}{\mathbf{3}}\right) \\ \mathbf{g}\mathbf{\text{'}}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{y}}^{\mathbf{-}\frac{\mathbf{2}}{\mathbf{3}}}}{\mathbf{3}} \\ \mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{y}}\mathbf{+}{\mathbf{C}}_{\mathbf{1}} \end{gathered}$$

Now$\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}{\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\sqrt[\mathbf{3}]{\mathbf{y}}\mathbf{+}{\mathbf{C}}_{\mathbf{1}}$

The general solution of the differential equation is${\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\sqrt[\mathbf{3}]{\mathbf{y}}\mathbf{=}\mathbf{C}$

Hence, the solution is role="math" localid="1664171463132" ${\mathbf{e}}^{\mathbf{x}}\mathbf{sin}\mathbf{y}\mathbf{-}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\sqrt[\mathbf{3}]{\mathbf{y}}\mathbf{=}\mathbf{C}$