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### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Decide whether the statement made is True or False. The relation ${\mathbf{sin}}{}{\mathbf{y}}{\mathbf{+}}{{\mathbf{e}}}^{{\mathbf{y}}}{\mathbf{=}}{{\mathbf{x}}}^{{\mathbf{6}}}{\mathbf{-}}{{\mathbf{x}}}^{{\mathbf{2}}}{\mathbf{+}}{\mathbf{x}}{\mathbf{+}}{\mathbf{1}}$ is an implicit solution to $\frac{\mathbf{dy}}{\mathbf{dx}}{\mathbf{=}}\frac{\mathbf{6}{\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{cos}\mathbf{y}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}}$.

The statement is true.

See the step by step solution

## Using the differential formula

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

## Differentiating  sin y+ey=x6-x2+x+1 with respect to x.

The Solution is given below,

$\frac{\mathbf{d}}{\mathbf{dx}}\left(\mathbf{sin}\mathbf{y}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}\right)\mathbf{=}\frac{\mathbf{d}}{\mathbf{dx}}\mathbf{\left(}{\mathbf{x}}^{\mathbf{6}}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\mathbf{cosy}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{6}{\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}\phantom{\rule{0ex}{0ex}}\mathbf{\left(}\mathbf{cosy}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}\mathbf{\right)}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{6}{\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}\phantom{\rule{0ex}{0ex}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{6}{\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}}{\mathbf{cosy}\mathbf{+}{\mathbf{e}}^{\mathbf{y}}}$

Hence, this is the given differential equation, the given statement is true.