Find a general solution of the system of equations dxdt 4x

Find a general solution of the system of equations: {dx/dt = 4x + 7y; dy/dt = x - 2y.

Solution

Adding the two equations gives

x\'+y\'=5(x+y)

ie

(x+y)\'=5(x+y)

ie

x+y=Ae^{5t}

x=Ae^{5t}-y

y\'=x-2y=Ae^{5t}-3y

y\'+3y=Ae^{5t}

INtegrating factor is: e^{3t}

Multiplying by integrating factor gives

(ye^{3t})\'=Ae^{8t}

Integrating gives

ye^{3t}=Ae^{8t}/8+B

y=Ae^{5t}/8+Be^{-3t}

x=Ae^{5t}-y=7Ae^{5t}/8-Be^{-3t}