# Math Help - Differential equation order3

1. ## Differential equation order3

Solve : y''' + y'' + y' = 8(x2+2x+2)

2. The DE ...

$y^{'''} + y^{''} + y^{'} = 8(x^{2} + 2x + 2)$ (1)

... has 'characteristic polynomial' ...

$t^{3} + t^{2} + t$ (2)

... and its roots are $t=0$ , $t= - \frac{1}{2} \pm i \frac{\sqrt{3}}{2}$. The 'general integral' of the 'incomplete equation' is then...

$y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x)$ (3)

It is easy to verify that $y= \frac{8}{3} x^{3}$ also satisfies (1) so that the general integral of (1) is...

$y= c_{1} + e^{-\frac{x}{2}} (c_{2} \cos \frac{\sqrt{3}}{2} x + c_{3} \sin \frac{\sqrt{3}}{2} x) + \frac{8}{3} x^{3}$ (4)

Kind regards

$\chi$ $\sigma$