Homogeneous Cauchy-Euler equation

Math 242 HW 7 (Sections 3.2 and 3.6)
Be sure to carefully justify all of your answers!
Section 3.2 Homogeneous equations with constant coefficients
Find the general solution to each differential equation.
1. x
(6) − 16x
00 = 0
2. x
(5) + 2x
000 + x
0 = 0
3. x
(4) − 5x
00 + 4x = 0
4. x
(5) − 5x
(4) + x
000 − 5x
00 = 0
5. x
(5) − 8x
00 = 0
1. x(t) = c1 + c2t + c3e
2t + c4e
−2t + c5 cos (2t) + c6 sin (2t)
2. x(t) = c1 + cost(c2 + c3t) + sin t(c4 + c5t)
3. x(t) = c1e
t + c2e
−t + c3e
2t + c4e
−2t
4. x(t) = c1 + c2t + c3e
5t + c4 cost + c5 sin t
5. x(t) = c1 + c2t + c3e
2t + e
−t

c4 cos 
t

3

+ c5 sin 
t

3

Section 3.6 Homogeneous Cauchy-Euler equation
Find the general solution to each differential equation:
1. 2t
2
d
2x
dt2
− 5t
dx
dt + 5x = 0
2. t
2
d
2x
dt2
+ t
dx
dt + 4x = 0
3. 4t
2
d
2x
dt2
+ 16t
dx
dt + 9x = 0
4. t
3
d
3x
dt3
+ t
dx
dt − x = 0
5. t
3
d
3x
dt3
− 3t
dx
dt + 3x = 0
1
6. t
3
d
3x
dt3
+ t
dx
dt − 2x = 0
1. x(t) = c1t + c2t
5/2
2. x(t) = c1 cos (2 ln t) + c2 sin (2 ln t)
3. x(t) = t
−3/2
(c1 + c2 ln t)
4. x(t) = c1 + c2ln t + c3 (ln t)
2
5. x(t) = c1t + c2t
3 + c3
1
t
6. x(t) = c1t
2 + c2t
1
2 cos √
3
2
ln t
!
+ c3t
1
2 sin √
3
2
ln t
!