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 2 -2 5] x. [x_1(t) x_2(t)] = c_1 [] + c_2 []

Solution

x_1\'=x_1-2x_2\\\\

x_2\'=2x_1+5x_2\\\\

Adding the two equations gives\\\\

x_1\'+x_2\'=3(x_1+x_2)

Integrating gives

x_1+x_2=c_1e^{3t}

x_1=c_1e^{3t}-x_2

x_2\'=2x_1+5x_2=2(c_1e^{3t}-x_2)+5x_2=3x_2+2c_1e^{3t}

x_2\'-3x_2=2c_1e^{3t}

Integrating factor is: e^{-3t}

Multiplying gives

(x_2\'-3x_2)e^{-3t}=2c_1

(x_2e^{-3t})\'=2c_1

Integrating gives

x_2e^{-3t}=2c_1t+c_2

x_2=2c_1te^{3t}+c_2e^{3t}

x_1=c_1e^{3t}-x_2=-c_1te^{3t}-c_2e^{3t}

x_1=-c_1te^{3t}-c_2e^{3t}