General solution to third order differential equation

[warfreak131]

y^III+y^II-y^I-y = 0

I used the characteristic equation and got:

r³+r²-r = 0

r (r²+r-1) = 0

Which means that r = 0 is one root,

And the other factors from the polynomial are (-1-Sqrt(5))/2 and (-1+Sqrt(5))/2

This means that the final answer would be:

y = C₁ Exp(0x) + C₂ Exp((-1-Sqrt(5))/2) + C₃Exp((-1+Sqrt(5))/2)

Then I'd simplify from there, but I checked my answer in the back of the book, and it says that I'm wrong. It says the answer is

y = C₁ Exp(x) + C₂ Exp(-x) + C₃x Exp(-x)

I also solved the equation using Mathematica and also got the answer that the book states.
Where did I go wrong?

 

[HallsofIvy]

y^III+y^II-y^I-y = 0

I used the characteristic equation and got:

r³+r²-r = 0

No, that's NOT the characteristic equation. That would be the characteristic equation for y'''+ y''- y'= 0. You seem to have forgotten the "- y" at the end.

The characteristic equation for y'''+ y''- y'- y= 0 is
$r^3+ r^2- r- 1= 0$
It should be easy to see that r= 1 is a root.

r (r²+r-1) = 0

Which means that r = 0 is one root,

And the other factors from the polynomial are (-1-Sqrt(5))/2 and (-1+Sqrt(5))/2

This means that the final answer would be:

y = C₁ Exp(0x) + C₂ Exp((-1-Sqrt(5))/2) + C₃Exp((-1+Sqrt(5))/2)

Then I'd simplify from there, but I checked my answer in the back of the book, and it says that I'm wrong. It says the answer is

y = C₁ Exp(x) + C₂ Exp(-x) + C₃x Exp(-x)

I also solved the equation using Mathematica and also got the answer that the book states.
Where did I go wrong?