Chapter 11, Problem 1RE

To determine

Whether $y=c{e}^{\frac{x}{6}}$ is the solution of the differential equation $6y^{\prime }=y$ or not.

Answer to Problem 1RE

Solution:

$y=c{e}^{\frac{x}{6}}$ is the solution of the differential equation $6y^{\prime }=y$.

Explanation of Solution

Given Information:

The provided differential equation is $6y^{\prime }=y$ and the provided solution is $y=c{e}^{\frac{x}{6}}$.

Consider the provided differential equation,

$6y^{\prime }=y$

The provided expression is $y=c{e}^{\frac{x}{6}}$.

Differentiate it with respect to x,

$\begin{array}{c}\frac{dy}{dx}=c\frac{d}{dx}\left(e^{\frac{x}{6}}\right)\\ y^{\prime }=\frac{c}{6}\left(e^{\frac{x}{6}}\right)\end{array}$

Substitute this value in the provided differential equation,

$\begin{array}{c}6y^{\prime }=y\\ 6\left(\frac{c}{6}\left(e^{\frac{x}{6}}\right)\right)=y\\ c{e}^{\frac{x}{6}}=y\\ y=y\end{array}$

$\text{LHS}=\text{RHS}$, hence verified.

Therefore, $y=c{e}^{\frac{x}{6}}$ is the solution of the differential equation $6y^{\prime }=y$.