Question

Find the general solution ?

Answer 1

See below.

Explanation:

Using the Laplace transform is handy

`X = ((x_1),(x_2))`
`A = ((1,1),(-4,1))`

`dot X = A X rArr sX(s)= A X(s)+x_0` then

`X(s) = (sI_2-A)^-1 x_0`

here

`(sI_2-A)^-1 =1/(s^2-2s+5) ((s-1,1),(-4,s-1))`

and `x_0 = (x_(10),x_(20))` are initial conditions.

then

`X(s) = 1/(s^2-2s+5)(((s-1)x_(10)+x_(20)),(-4x_(10)+(s-1)x_(20)))`

and

inverting

`X(t) = ((x_1(t)),(x_2(t)))=((1/2(2cos(2t)x_(10)+sin(2t)x_(20))),(cos(2t)x_(20)-2sin(2t)x_(10)))e^t u(t)`

here `u(t)` is the unitary step function.