Find the general solution ?

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1 Answer
Mar 20, 2018

Answer:

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.