1. (x(t) depends on the initial conditions)
Linear Differential Equations with Constant Coefficients:
(Characteristic polynomial) If
Different two roots
(No oscillation)
Repeated roots
Complex roots
(Oscillatory response)
The form of the system response is determined by the eigenvalues of the system which are determined by m, c and k.
Critical damping coefficient
(Damping ratio)
f(t) k c m
x(t)
Assume
Homogeneous Form
(x(t) depends on the initial conditions) 

2. Example:
f(t): Input
u(t): Output (response)
Homogeneous solution f(t)= u(t)=est
With Matlab:
Characteristic Equation:
s3est + 4s2est + 14sest + 20est= 0
a=[1,4,14,20];roots(a)
s3 + 4s2 + 14s + 20 = 0
Eigenvalues: -13i, - 2
uh(t) = C1e(-1+3i)t + C2 e(-1-3i)t + A2e-2t
uh(t) = A1e-tcos(3t-φ)+A2e-2t

3. uh(t) = A1e-tcos(3t-φ)+A2e-2t
Initial conditions: at t=0
-1.2 = A1 cosφ + A2
2.5 = -A1 cosφ +3A1 sinφ -2A2
-3.1= -8A1 cosφ - 6A1 sinφ + 4A2
A1, A2 and φ can be found by Newton-Raphson method.

4. Laplace Transform:

5. Laplace Transform of the Derivatives :

6. (shift in time or delay):

7. Laplace transform of the solution due to the initial conditions:
Initial conditions: at t=0

8. Partial fraction expansion:
With Matlab;
num=[-1.2,-2.3,-9.9]; den=[1,4,14,20]; [r,p,k]=residue(num,den)
r(1)= i, r(2)= i, r(3)=-1.01

9. Homogeneous solution :
uh(t) = A1e-tcos(3t-φ)+A2e-2t
With Matlab;
z= i
A1=2*abs(z)
fi=angle(z)

10. At t=0 are given. Find θ(t).
EXAMPLES:
The equation of the motion for the unforced motion of a simple pendulum is given as: m g θ
Joint friction, B L
m=2 kg
B=4 Nms/rad
L=2 m
At t= are given. Find θ(t).
Applying the Laplace transform,

11. [r,p,k]=residue(num,den)
EXAMPLES:
Laplace transform of the homogenous solution (due to the initial conditions)
Eigenvalues
The system is stable because the real parts of all the roots are negative.
clc;clear
num=[4 10];
den=[ ];
[r,p,k]=residue(num,den)
r(2)
A=2*abs(r(2))
Fi=angle(r(2)) Re
0.25
0.2556
Img

12. EXAMPLES:
clc;clear
dt=0.1418;
ts=25.149;
t=0:dt:ts;
tetat=0.7151*exp(-0.25*t).*cos(2.2006*t );
plot(t,tetat)