find the auxiliary equation and its roots, write a general solution of the ODE for: y\'\'\'-y\'\'-2y\'=0 and y\'\'+6y\'+10y=0 Solution Let y=emx y\' = memx, y\"= m2emx, y\"\'= m3emx substituting in the equation: m3emx - m2emx - memx = 0 taking emx common: emx (m3 - m2- 2m)= 0 (m3 - m2- 2m)= 0 Taking m common: m(m2 - m- 2)= 0 m(m2-2m+m-2) = 0 m(m+1)((m-2)=0 So we get three roots : m = 0 , -1 , 2 substituting in y = emx So,the general linear combination is the general solution of differential equation: y = c1e0x + c2e-x + c3e2x 2) y\'\'+6y\'+10y=0 m2emx + 6m +10emx = 0 emx (m2 + 6m +10) = 0 (m2 + 6m +10) = 0 using quadratic fromula to find the roots: m= -6 +- sqrt(36-40)/2 = -6 +- sqrt(-4)/2 = -3 +- i So, the general solution of differential equation in this case: y=erx( c1cos sx + c2sin sx) y= e-3x( c1cos x + c2sin x).

1. find the auxiliary equation and its roots, write a general solution of the ODE for:
y'''-y''-2y'=0
and
y''+6y'+10y=0
Solution
Let y=emx
y' = memx, y"= m2emx, y"'= m3emx
substituting in the equation:
m3emx - m2emx - memx = 0
taking emx common:
emx (m3 - m2- 2m)= 0
(m3 - m2- 2m)= 0
Taking m common:
m(m2 - m- 2)= 0
m(m2-2m+m-2) = 0
m(m+1)((m-2)=0
So we get three roots :
m = 0 , -1 , 2
substituting in y = emx
So,the general linear combination is the general solution of differential equation:
y = c1e0x + c2e-x + c3e2x
2)
y''+6y'+10y=0
m2emx + 6m +10emx = 0
emx (m2 + 6m +10) = 0
(m2 + 6m +10) = 0
using quadratic fromula to find the roots:
m= -6 +- sqrt(36-40)/2 = -6 +- sqrt(-4)/2 = -3 +- i
So, the general solution of differential equation in this case:
y=erx( c1cos sx + c2sin sx)
y= e-3x( c1cos x + c2sin x)