M427K Handout: Linear Systems: Systems of Linear First OrderODEs IISalman ButtApril 11, 2006ExerciseFind the general solution to the linear system~x0= 4 −38 −6!~xGraph TypesThe behavior of the vector field associated to a given linear system around the origin can bedescribed as follows:1.2.3.4.5.1Complex EigenvaluesRecall we have a 2 × 2 homogeneous linear system ~x0= A~x, and we guess our solution is~x = ~vertwhere ~v is some constant vector. The we are looking for eigenvalues r and eigenvectors~v. Now suppose our defining eigenvalue equation det(A − rI) = 0 yields two complex eigenvalues(in conjugate pairs) r1= λ + iµ, r2= λ − iµ. Then our corresponding constant vectors ~v1, ~v2willalso be complex conjugates. So we have two complex conjugate solutions ~x1= ~v1er1t, ~x2= ~v2er2t.To find real solutions, we merely take the real and imaginary parts of ~x1and ~x2(actually we justneed ~x1). We begin by writing ~v1as ~a + i~b, soMultiplying through we findFinally, we write ~x1(t) = ~u(t) + i~v(t) whereIt can be shown that ~u, ~v are linearly independent, so our final general solution becomesLet’s see an example:~x0= 1 −14 1!~x, ~x(0) =
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