Find the general solution ?

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Answer 1 · 2018-03-20T12:09:46

See below.

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.