Find the most general realvalued solution to the linear syst

Find the most general real-valued solution to the linear system of differential equations x\'_1 = 1 x_1 - 2 x_2, x\'_2 = 2 x_1 + 5 x_2 [x_1 (t) x_2 (t)] = c_1 [] + c_2 []

Solution

Adding two equatoins gives

x1\'+x2\'=3(x1+x2)

Integrating gives

x1+x2=Ae^{3t}

x2=Ae^{3t}-x1

x1\'=x1-2x2=-2Ae^{3t}+3x1

x1\'-3x1=-2Ae^{3t}

(x1\'-3x1)e^{-3t}=-2A

(x1 e^{-3t})\'=-2A

Integrating gives

x1 e^{-3t}=-2At+B

x1=e^{3t}(-2At+B)

x2=Ae^{3t}-x1=Ae^{3t}-e^{3t}(-2At+B)

x2=e^{3t}(2At+A-B)