Q9RP - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether the statement made is True or False. The relation ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{3}}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\mathbf{=}\mathbf{1}$  is an implicit solution to $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{y}}\mathbf{-}\mathbf{2}\mathbf{x}}{\mathbf{3}{\mathbf{y}}^{\mathbf{2}}}$ .

Step-by-Step Solution

Answer

The statement is false.

Using the differential formula

For the result use the differential formula  $\frac{\mathbf{d}}{\mathbf{dx}}\mathbf{(}{\mathbf{x}}^{\mathbf{n}}\mathbf{)}\mathbf{=}\mathbf{n}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$ and consider x and y as variable.

Differentiating x 2 + y 3 - e y = 1 with respect to x.

The given statement is,

$$\begin{gathered} {\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{3}}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\mathbf{=}\mathbf{1} \\ \frac{\mathbf{d}}{\mathbf{dx}}\mathbf{(}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{3}}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\mathbf{)}\mathbf{=}\frac{\mathbf{d}}{\mathbf{dx}}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}{\mathbf{y}}^{\mathbf{2}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{0} \\ \mathbf{(}\mathbf{3}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}{\mathbf{e}}^{\mathbf{y}}\mathbf{)}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{x} \\ \frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{2}\mathbf{x}}{{\mathbf{e}}^{\mathbf{y}}\mathbf{-}\mathbf{3}{\mathbf{y}}^{\mathbf{2}}} \end{gathered}$$

Since, this is the given differential equation, the given statement is false.