Question

For each of the following equations, find the most general function, so that the equation is exact.

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\left[\mathrm{ycos}\left(\mathrm{xy}\right)+e^x\right]\mathbf{dx}\mathbf{+}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{dy}\mathbf{=}\mathbf{0} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{(}\mathbf{y}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{3}}\mathbf{y}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{dx}\mathbf{+}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{dy}\mathbf{=}\mathbf{0} \end{gathered}$$

Short answer

$$\begin{gathered} \left(a\right)\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{\mathbf{sin}\mathbf{xy}}{\mathbf{y}}\mathbf{-}\mathbf{x}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{sin}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\mathbf{C} \\ \left(b\right)\mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{xy}}}{\mathbf{y}}\mathbf{+}{\mathbf{xe}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}{\mathbf{x}}^{\mathbf{4}}\mathbf{+}\mathbf{C} \end{gathered}$$

Step-by-step solution

Step 1(a): Find the value of∂N∂x

Here

$\left[\mathbf{y}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}\right]\mathbf{dx}\mathbf{+}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{dy}\mathbf{=}\mathbf{0}$

$\mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left[\mathbf{y}\mathbf{cos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}\right]$

Since the equation is exact.so, the condition for exact is$\frac{\partial \mathbf{M}}{\partial \mathbf{y}}=\frac{\partial \mathbf{N}}{\partial \mathbf{x}}$ .

$$\begin{gathered} \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\mathbf{=}\mathbf{-}\mathbf{xy}\mathbf{(}\mathbf{sin}\mathbf{xy}\mathbf{)}\mathbf{+}\mathbf{cos}\mathbf{xy}\mathbf{=}\frac{\partial \mathbf{N}}{\partial \mathbf{x}} \\ \frac{\partial \mathbf{N}}{\partial \mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{xy} \mathbf{(}\mathbf{sin} \mathbf{xy}\mathbf{)}\mathbf{+}\mathbf{cos} \mathbf{xy} ......\mathbf{\left(}\mathbf{1}\mathbf{\right)} \end{gathered}$$

Determine the value ofN (x,y) 

Integrating the equation (1), therefore

$\mathbf{N} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{\mathbf{sin} \mathbf{xy}}{\mathbf{y}}\mathbf{-}\mathbf{x} \mathbf{cos} \mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{sin} \mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\mathbf{C}$

.

Therefore The required function is $\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{\mathbf{sin}\mathbf{}\mathbf{xy}}{\mathbf{y}}\mathbf{-}\mathbf{xcos}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{sin}\mathbf{\left(}\mathbf{xy}\mathbf{\right)}\mathbf{+}\mathbf{C}$.

Step 3(b): Evaluate the value of ∂N∂x

Here$$\begin{gathered} \mathbf{(}\mathbf{y}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{3}}\mathbf{y}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{dx}\mathbf{+}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{dy}\mathbf{=}\mathbf{0} \\ \mathbf{M}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{y}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{3}}\mathbf{y}\mathbf{+}\mathbf{2}\mathbf{)} \end{gathered}$$

Since the equation is exact. So, the condition for exact is$\frac{\partial \mathbf{M}}{\partial \mathbf{y}}=\frac{\partial \mathbf{N}}{\partial \mathbf{x}}$ .

$$\begin{gathered} \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\mathbf{=}{\mathbf{e}}^{\mathbf{xy}}\mathbf{+}\mathbf{y}{\mathbf{xe}}^{\mathbf{xy}}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{3}}\mathbf{=}\frac{\partial \mathbf{N}}{\partial \mathbf{x}} \\ \frac{\partial \mathbf{N}}{\partial \mathbf{x}}\mathbf{=}{\mathbf{e}}^{\mathbf{xy}}\mathbf{+}\mathbf{y}{\mathbf{xe}}^{\mathbf{xy}}\mathbf{-}\mathbf{4}{\mathbf{x}}^{\mathbf{3}} ......\mathbf{\left(}\mathbf{2}\mathbf{\right)} \end{gathered}$$

Determine the value of N.(x,y)     

Integrating the equation (2) therefore

$\text{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{xy}}}{\mathbf{y}}\mathbf{+}{\mathbf{xe}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}{\mathbf{x}}^{\mathbf{4}}\mathbf{+}\mathbf{C}$

.

Therefore, the required function is$\text{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{xy}}}{\mathbf{y}}\mathbf{+}{\mathbf{xe}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}{\mathbf{x}}^{\mathbf{4}}\mathbf{+}\mathbf{C}$