Q8RP - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Decide whether the statement made is True or False. The function  $\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}$ is a solution to $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}}$.

Step-by-Step Solution

Answer

The statement is false.

Differentiating y ( x ) = - 1 3 ( x + 1 ) concerning x.

The given differential equation is,

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

Substituting the value of y(x) in dy dx = y - 1 x + 3 .

Right hand side of the equation becomes,

$$\begin{gathered} \frac{\mathbf{y}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}}\mathbf{=}\frac{\mathbf{-}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}\mathbf{1}}{\mathbf{x}\mathbf{+}\mathbf{3}} \\ \mathbf{=}\frac{\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{3}}{\mathbf{3}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{)}} \\ \mathbf{=}\frac{\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{4}}{\mathbf{3}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{)}} \end{gathered}$$

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.