Chapter 4 - Higher Order Linear Equations - 4.2 Homogenous Equations with Constant Coefficients - Problems - Page 232: 12

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

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