Solve : y''' + y'' + y' = 8(x2+2x+2)
The DE ...
$\displaystyle y^{'''} + y^{''} + y^{'} = 8(x^{2} + 2x + 2)$ (1)
... has 'characteristic polynomial' ...
$\displaystyle t^{3} + t^{2} + t$ (2)
... and its roots are $\displaystyle t=0$ , $\displaystyle t= - \frac{1}{2} \pm i \frac{\sqrt{3}}{2}$. The 'general integral' of the 'incomplete equation' is then...
$\displaystyle y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x)$ (3)
It is easy to verify that $\displaystyle y= \frac{8}{3} x^{3}$ also satisfies (1) so that the general integral of (1) is...
$\displaystyle y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x) + \frac{8}{3} x^{3} $ (4)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$