Chapter 3 - Second Order Linear Equations - 3.1 Homogenous Equations with Constant Coefficients - Problems - Page 144: 2

$${y}=c_{1}\exp^{-2x}+c_{2}\exp^{-x}$$

$$y''+3y'+2y=0$$Let $y-e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$. $${\lambda}^2+3{\lambda}+2=0$$ $$(\lambda+2)(\lambda+1)=0$$ $$\lambda_{1,2}=-2,-1$$ $$\therefore{y}=c_{1}\exp^{-2x}+c_{2}\exp^{-x}$$