Find the general solution of the system of ODE?

x' = 2x + y

y' = x + 2y

The fancy way to do it is by recasting it in terms of matrices:

write X = [x; y], A = [2, 1; 1, 2], then

X' = AX

a solution is of the form X = B exp(rt), where exp is the exponential, r is to be determined as well as the constant vector B.

X = B exp(rt), X' = r B exp(rt), insert this into the differential equation

X' = AX

rB exp(rt) = A B exp(rt)

rB exp(rt) = AB exp(rt)

rIB exp(rt) = AB exp(rt)

or (A - rI)B exp(rt) = 0

since exp(rt) =/= 0, then

(A - rI)B = 0

thus we see that B satisfies an eigen equation whose eigenvalues are given by r.

nontrivial solutions (cf. a course linear algebra to know why) would correspond to

det(A - rI) = 0

find r, call them r1 and r2 with corresponding eigenvectors B1 and B2 (find these)

Your final solutions is X = C (B1 exp(r1 t)) + D (B2 exp(r2 t)) ......... [Ans.]