Q11 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 9–13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

${\mathbf{e}}^{\mathbf{xy}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{1}$, $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{-}\mathbf{y}}{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{+}\mathbf{x}}$

Step-by-Step Solution

Answer

Yes, the given relation is an implicit solution to the given differential equation.

Step 1: Differentiate the given relation

As, in the given relation ${\mathrm{e}}^{\mathrm{xy}}+\mathrm{y}=\mathrm{x}-1$, y is defined implicitly as the function of x, so by using implicit differentiation, we will differentiate the given relation concerning x,

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{xy}}+\mathrm{y}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}-1\right) \\ {\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{y}+\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\frac{\mathrm{dy}}{\mathrm{dx}}=1 \\ \frac{\mathrm{dy}}{\mathrm{dx}}[\mathrm{x}{\mathrm{e}}^{\mathrm{xy}}+1]=1-\mathrm{y}{\mathrm{e}}^{\mathrm{xy}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1-{\mathrm{ye}}^{\mathrm{xy}}}{1+{\mathrm{xe}}^{\mathrm{xy}}} \end{gathered}$$

Step 2: Simplification of the differential equation

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1-{\mathrm{ye}}^{\mathrm{xy}}}{1+{\mathrm{xe}}^{\mathrm{xy}}}\times \frac{{\mathrm{e}}^{-\mathrm{xy}}}{{\mathrm{e}}^{-\mathrm{xy}}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{e}}^{-\mathrm{xy}}-\mathrm{y}}{{\mathrm{e}}^{-\mathrm{xy}}+\mathrm{x}} \end{gathered}$$

Which is identical to the given differential equation.

Thus, the relation ${\mathit{e}}^{\mathbf{x}\mathbf{y}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{,}$ is an implicit solution to the differential equation $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{-}\mathbf{y}}{{\mathbf{e}}^{\mathbf{-}\mathbf{xy}}\mathbf{+}\mathbf{x}}$.