Answer
$y= C_{1}e^{\frac{-1}{2}t}+\;[C_{2}e^{\frac{-1}{3} t}cos(\frac{\sqrt{3}}{3} t)+C_{3}e^{\frac{-1}{3} t}sin(\frac{\sqrt{3}}{3} t)\;]$
Work Step by Step
Let $\;\;\;\;\;y=e^{rt}\\\\$
$18{y}'''+21{y}''+14{y}'+4y=0 \;\;\;\;\Rightarrow \;\;\;\; 18r^3e^{rt}+21r^2e^{rt}+14re^{rt}+4e^{rt}=0\\\\$
$18r^3+21r^2+14r+4=(2r+1)(9r^2+6r+4)=0 $$\rightarrow \;\;\;\;\;\; r_{1}= \frac{-1}{2}\;\;\;\;\;or\;\;\;\;\;\;r_{2}=\frac{-1}{3}-i\frac{\sqrt{3}}{3}\;,\;r_{3}=\frac{-1}{3}+i\frac{\sqrt{3}}{3} \;\;\;\;\;\;\\\\$
So the 3 roots are: $\;\;\;r_{1}=\frac{-1}{2} \;\;\;\;\; ,\;\;\;\;r_{2},r_{3}=\frac{-1}{3}\pm i\frac{\sqrt{3}}{3} $
The general solution for complex roots is:
$y= C_{1}e^{\alpha t}cos(\beta t)+C_{2}e^{\alpha t}sin(\beta t)$
$y= C_{1}e^{\frac{-1}{2}t}+\;[C_{2}e^{\frac{-1}{3} t}cos(\frac{\sqrt{3}}{3} t)+C_{3}e^{\frac{-1}{3} t}sin(\frac{\sqrt{3}}{3} t)\;]$
