Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems - Page 21: 23

$r=-3$

$y''+6y'+9y=0$ ______($\alpha$) $y=e^{rx}$ ______(1) Differentiate (1) with respect to $x$ $y'=re^{rx}$ ______(2) Differentiate (2) with respect to $x$ $y''=r^2e^{rx}$ ______(3) Since $y=e^{rx}$ is solution of ($\alpha$) so it satisfies ($\alpha$) $r^2e^{rx}+6re^{rx}+9e^{rx}=0$ $e^{rx}[r^2+6r+9]=0$ $e^{rx}\neq 0$ Therefore, $r^2+6r+9=0$ $(r+3)^2=0$ $r=-3$ Hence, $r=-3$