Find the general solution to the system dx/dt=Ax where A[0,1;1,0]
That reduce to two coupled equations
dx1/dt = x2...(1)
dx2/dt = x1...(2)
Differentiate (1),
d²x1/dt²
= dx2/dt
= x1
d²x1/dt² -x1=0 ...(3)
Solve (3) to get
x1=C1e^t+C2e^(-t) ...(4)
Substitute (4) in (2) to get
dx2/dt = C1e^t+C2e^(-t)
Integrate
x2 = C1e^t-C2e^(-t) ....(5)
So the solution is:
x1=C1e^t+C2e^(-t) ...(4)
x2 = C1e^t-C2e^(-t) ....(5)
Backsubstitute in (1) and (2) to make sure they work.