Question

In Problems 21–26, solve the initial value problem.

$\left({\mathbf{ye}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\right)\mathbf{dx}\mathbf{+}\left({\mathbf{xe}}^{\mathbf{xy}}\mathbf{+}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right)\mathbf{dy}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$

Short answer

${\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{=}\mathbf{e}\mathbf{-}\mathbf{1}$

Step-by-step solution

Evaluate the equation is exact

Here$\left({\mathbf{ye}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\right)\mathbf{dx}\mathbf{+}\left({\mathbf{xe}}^{\mathbf{xy}}\mathbf{+}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right)\mathbf{dy}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$

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

$$\begin{gathered} \mathbf{M} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left({\mathbf{ye}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\right) \\ \mathbf{N} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left({\mathbf{xe}}^{\mathbf{xy}}\mathbf{+}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right) \\ \frac{\partial \mathbf{M}}{\partial \mathbf{y}}\mathbf{=}{\mathbf{e}}^{\mathbf{xy}}\mathbf{(} \mathbf{xy}\mathbf{+}\mathbf{1} \mathbf{)}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{=}\frac{\partial \mathbf{N}}{\partial \mathbf{x}} \end{gathered}$$

Therefore this equation is exact.

Find the value of F (x,y) 

Here

$$\begin{gathered} \mathbf{M} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\left({\mathbf{ye}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\right) \\ \mathbf{F} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\int \mathbf{M} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \mathbf{dx}\mathbf{+}\mathbf{g} \mathbf{\left(}\mathbf{y}\mathbf{\right)} \\ \mathbf{=}\int \left({\mathbf{ye}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{y}}\right) \mathbf{dx}\mathbf{+}\mathbf{g} \mathbf{\left(}\mathbf{y}\mathbf{\right)} \\ \mathbf{=}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{+}\mathbf{g} \mathbf{\left(}\mathbf{y}\mathbf{\right)} \end{gathered}$$

Determine the value of g(y)

$$\begin{gathered} \frac{\partial \mathbf{F}}{\partial \mathbf{y}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{N}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \\ \left({\mathbf{xe}}^{\mathbf{xy}}\mathbf{+}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right)\mathbf{+}\mathbf{g}\mathbf{\text{'}}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\left({\mathbf{xe}}^{\mathbf{xy}}\mathbf{+}\frac{\mathbf{x}}{{\mathbf{y}}^{\mathbf{2}}}\right) \\ \mathbf{g}\mathbf{\text{'}}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\mathbf{0} \\ \mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{C}}_{\mathbf{1}} \end{gathered}$$

Now$\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{=}{\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{+}{\mathbf{C}}_{\mathbf{1}}$

Therefore the solution of the differential equation is${\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{=}\mathbf{C}$

.

Apply the initial conditions.$\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$

$$\begin{gathered} {\mathbf{e}}^{\mathbf{1}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{=}\mathbf{C} \\ \mathbf{C}\mathbf{=}\mathbf{e}\mathbf{-}\mathbf{1} \end{gathered}$$

Therefore, the solutions is role="math" localid="1664180725870" ${\mathbf{e}}^{\mathbf{xy}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{=}\mathbf{e}\mathbf{-}\mathbf{1}$.