Solve y Ayt y 5 3 6 4ytSolutionLet yuvT u5u3v v6u4v uvuv o

Solve y\' = Ay(t): y\' = [5 3 -6 -4]y(t)

Solution

Let, y=(u,v)^T

u\'=5u+3v

v\'=-6u-4v

u\'+v\'=-u-v

or, (u+v)\'=-(u+v)

Integrating gives:

u+v=Ae^{-t}

2u\'+v\'=10u+6v-6u-4v=4u+2v=2(2u+v)

Or (2u+v)\'=2(2u+v)

Integrating gives:

2u+v=Be^{2t}

u+v=Ae^{-t}

Hence, u=Be^{2t}-Ae^{-t}

v=2Ae^{-t}-Be^{2t}

This solves the system.