Question

Derive the solution of homogeneous state equations​

Answer 1

Answer:

A first order differential equation is said to be homogeneous if it may be written. where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form.

Answer 2

Answer:Taking Laplace transform of Eq. (9.55), we get,

 

sX(s) − x(0) = AX(s)

 

∴             (sI – A) X(s) = x(0)

 

∴          X (s) = (sI – A)−1x(0)       (9.56)

 

∴          X(s) = ϕ(s) x (0)       (9.57)

 

where             ϕ(s) = (sI – A) −1       (9.58)

 

called the resolvent matrix.

 

image

 

Taking inverse Laplace transform of Eq. (9.59), it can be written as

 

x(t) = ϕ(t) x(0)       (9.60)

 

where ϕ(t) is given in Eq. (9.60) ...