MATH30650 Lecture Notes - Lecture 13: Joule, Eigenvalues And Eigenvectors, Integrating Factor
Document Summary
In these notes, we describe a method for solving nonhomogeneous systems of di eren- tial equations ((cid:133)rst order or second order). This method is equivalent to the diagonaliza- tion method in the book, but is more computationally manageable. The idea is to express our solutions in terms of the eigenvectors of a: since a applied to an eigenvector is a multiple of that eigenvector, we get separate (uncoupled) equations for each coe cient. Thus, this method has the e ect of turning a system of coupled equations into a system of uncoupled equations that can be solved one at a time by the methods of chapter 3. 0 = ay + g(t); where a is a constant n(cid:2)n matrix and g is a vector-valued function. We now guess a particular solution y to our equation of the form y(t) = y1(t)v1 + y2(t)v2 + (cid:1) (cid:1) (cid:1) + yn(t)vn: (2) Let us now plug this back into our equation.
