1. SYSTEM OF DIFFERENTIAL EQUATIONS
f(t) : Input
u(t) and v(t) : Outputs to be found
System of constant coefficient differential equations with two unknowns
* First order derivative terms are on the left hand side
x : State variable matrix (nx1)
A : System matrix (nxn)
* Non-derivative terms are on the right hand side
u : Input matrix (mx1)
B : Matrix with dimension (nxm)

2. s I X(s) – x0 = A X(s) + B U (s)
Laplace Transformation:
s X(s) - x0 = A X(s) + B U(s)
s I X(s) – x0 = A X(s) + B U (s)
[s I – A] X(s) = x0 + B U (s)
X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)
Homogenous part
Particular part

3. Example:
At t=0 u=-2 , v=3 ; F(s) = -4/s
D(s)=det[sI-A]=s2+3s+120
: Eigenvalue equation
X(s) = [s I – A]-1 x0 + [s I – A]-1 B U (s)

4. With Matlab :
Calculation of eigenvalues:
a=[0,1 ; -120,-3] ; eig(a)
Determination of matrix [sI-A]-1:
syms s;
a=[0,1;-120,-3] ;
i1=eye(2);
a1=inv(s*i1-a);
pretty(a1)

5. Example:
f(t):Input
x(t) and y(t) : Outputs to be found
and
Let’s use the variables