Let xt x1 t x2 t be a solution to the system of differentia

Let x(t) = [x_1 (t) x_2 (t)] be a solution to the system of differential equations: x_1\' (t) = -5 x_1 (t) - 2x_2 (t) x_2\' (t) = 3 x_1 (t) If x(0) = [3 4], find x (t). Put the eigenvalues in ascending order when you enter x_1(t), x_2(t) below.

Solution

Differentiation first ODE gives

x1\'\'=-5x1\'-2x2\'=-5x1\'-6x1

x1\'\'+5x1\'+6x1=0

Assume exponential solution: x1= e^{kt} and substitutte giving

k^2+5k+6=0

k=-2,-3

So,

x1= A exp(-2t)+B exp(-3t)

x2\'=3x1=3 A exp(-2t)+3B exp(-3t)

INtegrating gives

x2=-3A/2 exp(-2t)- B exp(-3t)

Using initial conditions:

x1(0)=3, x2(0)=4

So,

A+B=3, -3A/2-B=4

So, -A/2=7 ie A=-14

and B =17

So,

x1= -14 exp(-2t)+17 exp(-3t)

x2=21 exp(-2t)- 17 exp(-3t)