Chapter 8 - Linear Differential Equations of Order n - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - Problems - Page 513: 12

$y(x)=C_1e^{-2x}+C_2 xe^{-2x}$

Solve the characteristic equation for the differential equation. $$(D+2)^2 y=0$$ Factor and solve for the roots. $$(r+2)^2=0$$ $$r_1=-2, r_2=-2$$ This implies that there are two independent solutions to the differential equation $y_1(x)=e^{-2x}$ and $y_2=xe^{-2x}$. Therefore, the general equation is equal to $y(x)=C_1e^{-2x}+C_2 xe^{-2x}$.