Chapter 31, Problem 1RE

To determine

Whether the statement “$y=c{e}^{-x}+2e^{2x}$ is a solution of the differential equation $D^{2}y-Dy=2y$” is true or false.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Definition used:

“A solution of a differential equation is a relation between the variables that satisfies the differential equation. That is, when this relation is substituted into the differential equation, an algebraic identity results”.

Calculation:

The given differential equation is $D^{2}y-Dy=2y$ and the function is $y=c{e}^{-x}+2e^{2x}$.

Obtain the first derivative of $y=c{e}^{-x}+2e^{2x}$ as follows.

$\begin{array}{c}y=c{e}^{-x}+2e^{2x}\\ Dy=c{e}^{-x}\left(-1\right)+2e^{2x}\left(2\right)\\ Dy=-c{e}^{-x}+4e^{2x}\end{array}$

Obtain the second derivative of $y=e^x+e^{-x}$ as follows.

$\begin{array}{c}{D}^{2}y=-c{e}^{-x}\left(-1\right)+4e^{2x}\left(2\right)\\ =c{e}^{-x}+8e^{2x}\end{array}$

Substitute values of $D^{2}y,Dy\text{ and }y$ in $D^{2}y-Dy=2y$ as follows.

$\begin{array}{l}c{e}^{-x}+8e^{2x}-\left(-c{e}^{-x}+4e^{2x}\right)=2\left(c{e}^{-x}+2e^{2x}\right)\\ c{e}^{-x}+8e^{2x}+c{e}^{-x}-4e^{2x}=2c{e}^{-x}+4e^{2x}\\ 2c{e}^{-x}+4e^{2x}=2c{e}^{-x}+4e^{2x}\end{array}$

It is observed that, the equation $y=e^x+e^{-x}$ satisfies the differential equation $D^{2}y-Dy=2y$.

Therefore, the given statement is true.