Chapter 8 - Linear Differential Equations of Order n - 8.2 Constant Coefficient Homogeneous Linear Differential Equations - Problems - Page 514: 26

$y(x)=C_1+C_2x+C_3e^x$

Solve the auxiliary equation for the differential equation. $$r^2(r-1)=0$$ Factor and solve for the roots. Roots are: $r_1=0$, as a multiplicity of $2$ and $r_2=1$ as a multiplicity of $1$. This implies that there are two independent solutions to the differential equation $y_1(x)=1$ and $y_2=x$ and $y_3=e^x$ Therefore, the general equation is equal to $y(x)=C_1+C_2x+C_3e^x$