Chapter 3 - Second Order Linear Equations - 3.1 Homogenous Equations with Constant Coefficients - Problems - Page 144: 6

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

$$4y''-9y=0$$Let $y-e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$. $$4{\lambda}^2-9=0$$ $$\lambda_{1,2}=\pm\frac{3}{2}$$ $$\therefore{y}=c_{1}\exp^{\frac{3}{2}x}+c_{2}\exp^{\frac{-3}{2}x}$$