Chapter 4 - Higher Order Linear Equations - 4.2 Homogenous Equations with Constant Coefficients - Problems - Page 232: 16

$y=C_{1}e^{2t}+C_{2} e^{-2t}+C_{3}e^{t} + C_{4}e^{-t}$

Let $\;\;\;\;y=e^{rt}\\\\$ $y^{(4)}-5{y}''+4y=0 \;\;\;\; \Rightarrow \;\;\;\;r^4e^{rt}-5r^2e^{rt}+4e^{rt}=0\\\\$ $r^4-5r^2+4=(r^2-4)(r^2-1)=0 \;\;\;\;\rightarrow \;\;\;\;r_{1}=2,r_{2}=-2\;\;\;\;or\;\;\;\; r_{3}=1,r_{4}=-1\\\\$ $y=C_{1}e^{2t}+C_{2} e^{-2t}+C_{3}e^{t} + C_{4}e^{-t}$