Chapter 3 - Second Order Linear Equations - 3.1 Homogenous Equations with Constant Coefficients - Problems - Page 144: 8

$${y}=c_{1}\exp^{1+\sqrt{3}x}+c_{2}\exp^{1-\sqrt{3}x}$$

$$y''-2y'+2y=0$$Let $y-e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$. $${\lambda}^2-2{\lambda}+2=0$$ $$\lambda_{1,2}=\frac{2\pm{\sqrt{4+8}}}{2}=\frac{2\pm{2\sqrt3}}{2}=1\pm\sqrt3$$ $$\therefore{y}=c_{1}\exp^{1+\sqrt{3}x}+c_{2}\exp^{1-\sqrt{3}x}$$