M427K Handout: Linear Systems: Systems of Linear First OrderODEs IIISalman ButtApril 13, 2006Notes1. Office hours this Monday are abbreviated to 10-10:30am.Complex EigenvaluesLast time, we left off consider the linear system~x0= 1 −14 1!~x, ~x(0) = 1−1!We found the matrix had complex eigenvalues 1 ± 2i with eigenvector ~v = (1, −2i). We wrote oursolution as ~x1(t) = ~ve(1+2i)t, determined its real and imaginary parts and wrote down our generalsolution:~x(t) = c1et cos 2t2 sin 2t!+ c2et sin 2t−2 cos 2t!Let’s determine our constants c1, c2:ExerciseFind the solution to the linear system~x0= 0 1−2 2!~x, ~x(0) = 21!1Exponential of a MatrixRecall that when we have the first-order differential equation x0= ax, the solution was x(t) =x(0)eat. We use this as inspiration to guess that a possible solution to the differential equation~x0= A~x is ~x = ~x(0)eAt. But what is eAt, or more simply, what is eA? We define it as follows:Thus eAtis merelySo if want to plug this guess into our differential equation, we nee d to compute the derivativeof eAt:Writing our differential equation as ~x0− A~x = 0, we haveSo if we can compute eAtquickly, we can find our solution in no time. And when can we2compute eAtquickly? Let’s consider the matrices: 2 10 2! 2 00 4!These examples hold true in general: if A is upper (lower) triangular or is diagonal, determiningeAtis easy.Let’s actually solve a differential equation using this technique:~x0= 2 01 2!~x, ~x(0) = 1−1!Repeated EigenvaluesIn this section, we consider what happens when our homogeneous system ~x0= A~x has repeatedeigenvalues r1= r2. As before we compute the eigenvector ~v which is now associated to both r1and r2. This will give us one solution ~x1= ~vert. But what about a second solution? Well, it turnsout that the second solution will have the form ~x2= ~vtert+ ~uertwhere ~v is the eigenvector for rabove and ~u satisfies the equation(A − rI)~u = ~v.Such a vector ~u is c alled a generalized eigenvector associated to r. Finally we set the generalsolution to be x(t) = c1~x1(t) + c2~x2(t).3Let’s see an example:~x0= 1 −11
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