Summary MAT3706 Full study notes 2022

MAT3706 Full study notes 2022 Chapter(1( ! ! x1 (t) = a11(t) x1 + a12 (t) x2 + :… + a1n(t) xn + Q1(t) - Qk(t) are called forcing terms - homogeneous when Qk (t) is zero for all k = 1; 2; … ; n - Non-homogenous if at least one Q k (t) is not zero - Solution is defined and differentiable Representation of homogenous system: Higher–order system with constant coefficients - Has polynomial differential operators Determinant of the system: Degenerate: If det = 0 Non–degenerate: If det ≠ 0 Theorem 1.4 (The Superposition Principle): Any linear combination of solutions of a system of differential equations is also a solution of the system. ! We say that X1 (t) ; X2 (t) ; … ;Xn (t) are linearly independent if it follows from c1X1 + c2X2 + … + c nXn = 0 for all t in J, that c 1 = c2 = … = c n = 0. n solutions are linearly independent if the Wronskian determinant is never zero General solution (HOMOGENOUS): The sum of all multiples of linearly independent solutions. ! ! General solution (NON-HOMOGENOUS): The sum of all multiples of linearly independent solutions, if [x1p (t) ; x2p (t) ; : : : ; xnp (t)]T is a particular solution of the inhomogeneous system