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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 function $\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\right)}$ is a solution to $\frac{\mathbf{dy}}{\mathbf{dx}}{\mathbf{=}}\frac{\mathbf{y}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}}$.

The statement is false.

See the step by step solution

## Differentiating y(x)=-13(x+1) concerning x.

The given differential equation is,

$\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{\left(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}$

## Substituting the value of y(x) in  dydx=y-1x+3.

Right hand side of the equation becomes,

$\frac{\mathbf{y}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}}\mathbf{=}\frac{\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\right)}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}}\phantom{\rule{0ex}{0ex}}\mathbf{=}\frac{\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{3}}{\mathbf{3}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{\right)}}\phantom{\rule{0ex}{0ex}}\mathbf{=}\frac{\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{4}}{\mathbf{3}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{\right)}}$

Left Hand Side of the equation becomes,

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}$

LHS ≠ RHS

Hence, the statement is not true.