question of 3rd order differential equation:y'''+y=0?

Answer 1

The characteristic equation is
r^3+1=0
(r+1)(r^2-r+1)=0,with roots
r=-1
r=[1+isqrt(3)]/2
r=[1-isqrt(3)]/2
So,
y(x)=c1e^(-x)+
c2e^[(1/2+isqrt(3)/2)x]
+c3e^[(1/2-isqrt(3)/2)x]
y(x)=c1e^(-x)+
c2{[e^[(1/2)x]+
e^[(isqrt(3)/2)x]}
+c3{[e^[(1/2)x]+
e^[(-isqrt(3)/2)x]}
Using the formula
e^(ix)=cosx+isinx,we take
y(x)=c1*e^(-x)+
c2'*e^[(1/2)x]*
cos[sqrt(3)/2)x]
+c3'*e^[(1/2)x]*
sin[(sqrt(3)/2)x]

Answer 2

The solutions are e^-x
xe^-x and x^2e^-x as the caracteristic equation is r^3+1=0

y=C-1e^-x +C-2 xe^-x +C_3 x^2e^-x where the C´s are arbitrary constants

Answer 3

I'm pretty sure the answer you're looking for is Ce^(-x). Actually, to be even more defined, it could be Ce^(-x+b)...if im not mistaken.