Find the solution to the linear system of differential equat

Find the solution to the linear system of differential equations {x\' = -16x + 18y y\' = -12x + 14y satisfying the initial conditions x(0) = 6 and y(0) = 3. x(t) = y(t) =

Solution

Subtracting two equations gives

x\'-y\'=-4(x-y)

(x-y)\'=-4(x-y)

Integrating

x-y=Ae^{-4t}

x=y+Ae^{-4t}

y\'=-12x+14y=2y-12Ae^{-4t}

(y\'-2y)=-12Ae^{-4t}

(y\'-2y)e^{-2t}=-12Ae^{-6t}

INtegrating gives

ye^{-2t}=2Ae^{-6t}+B

y=2Ae^{-4t}+Be^{2t}

x=y+Ae^{-4t}

x=3Ae^{-4t}+Be^{2t}

Usingin initial conditions

3A+B=6

2A+B=3

So, A=3,B=-3