Find the real form of the general solution for the system of

Find the real form of the general solution for the system of differential equations x\' = (1 -2 4 -3)x.

Solution

LEt, x=[u v]^T

u\'=u-2v

v\'=4u-3v

v=(u-u\')/2

v\'=(u\'-u\'\')/2=4u-3v=4u-3(u-u\')/2

(u\'-u\'\')/2=(8u-3u+3u\')/2

u\'-u\'\'=5u+3u\'

u\'\'+2u\'+5u=0

Assume, u=exp(kt)

Substituting gives

k^2+2k+5=0

k=-1+2i,-1-2i

So, u=exp(-t)(A sin(2t)+B cos(2t))

v=(u-u\')/2

u\'=-u+exp(-t)(2A cos(2t)-2B sin(2t))

v=(2u-exp(-t)(2A cos(2t)-2B sin(2t)))/2

v=exp(-t)(A sin(2t)+B cos(2t))-exp(-t)(A cos(2t)-B sin(2t))