Chapter 4 - Higher Order Linear Equations - 4.2 Homogenous Equations with Constant Coefficients - Problems - Page 232: 11

$y(t)=C_{1}e^t+C_{2}te^t+C_{3}e^{-t}$

let : $\;\;\;\;\;\;y=e^{rt}$ ${y}'=re^{rt}\;\;\;\;\;\;\;\;\;{y}''=r^2e^{rt}\;\;\;\;\;\;\;\;{y}'''=r^3e^{rt}\\\\$ $(r^3-r^2-r+1)e^{rt}=0\\\\$ $\Rightarrow r^3-r^2-r+1=0\\\\$ $r(r^2-1)-(r^2-1)=0\;\;\;\;\;\;\rightarrow \;\;\;\;\;\;(r^2-1)(r-1)=0\\\\$ $r^2=1\;\;\;\;\;\;\;\;\Rightarrow \;\;\;r_{1}= 1,r_{2}=-1\;\;\;\;,\;\;\;\; \;\;r_{3}=1$ $y(t)=C_{1}e^t+C_{2}te^t+C_{3}e^{-t}$