Lec 30 MIT 18.03 Differential Equations, Spring 2006
Everything I say today is going to be for n-by-n systems, but for your calculations and the exams two-by-two will be good enough. Our system looks like that. Notice I am talking today about the homogeneous system, not the inhomogenous system. So, homogenous. And we have so far two basic methods of solving it. The first one, on which we spent the most time, is the method of where you calculate the eigenvalues of the matrix, the eigenvectors, and put them together to make the general solution. So eigenvalues, e-vectors and so on. The second method, which I gave you last time, I called "royal road," simply calculates the matrix e to the At and says that the solution is e to the At times x zero, the initial condition. That is very elegant. The only problem is that to calculate the matrix e to the At, although sometimes you can do it by its definition as an infinite series, most of the time the only way to calculate the matrix e to the At is by using the fundamental ...