Chapter 8 - Linear Differential Equations of Order n - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - Problems - Page 513: 9

$y=C_1e^{3t}+C_2te^{3t}$

Solve the characteristic equation for the differential equation. $$r^2-6r-9=0$$ Factor and solve for the roots. $$(r-3)(r-3)=0$$ $$r=3,3$$ The general equation for repeated roots is equal to $y=C_1e^{r_1t}+C_2te^{r_1t}$. Therefore, the solution equals $y=C_1e^{3t}+C_2te^{3t}$.