Summary MAT3706 - Ordinary Differential Equations Full_study_notes.pdf

MAT3706 - Ordinary Differential Equations Full_study_ 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 Qk (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 + … + cnXn = 0 for all t in J, that c1 = c2 = … = cn = 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. Order(of(determinant =!the correct number of arbitrary constants in a general solution of a system of differential equations is equal to the order of the determinant of the system - Theorem 1.9 Δ = det(A) If Δ is identically zero, then the system either has either infinitely many solutions, or no solutions. Existence and Uniqueness Theorem: Theorem 1.10 - A first order partial differential equation with an IVP has a unique solution. Method of elimination = operator method - used for non-degenerate systems Polynomial operator: P (D) - P(D)|y| Linearity property: Closed under addition and multiplication Product of polynomial differential operators Differential operators can be treated just like numbers: IE D2 – 1 = (D+1)(D+1) Elimination(method((Operator(method) 1)!Eliminate either!x!or!y 2 To!eliminate!x,!times!first!equation!by!coefficient!of!x in!second!equation,!and! times!the!second!equation!by!the!coefficient!of!x in!the!first!equation. 2 To!eliminate!y,!repeat!above!but!with!y!coefficient. 2)!Add!or!subtract!the!two!equations!with!each!other!to!eliminate!either!the!x!or!the!y! variable. 3)!Find!the!auxiliary!equation of!the!equation!left!over:!Take!all!the!derivatives!to!the! left,!make!it!equal!0,!and!then!sub!‘m’!in!place!of!D,!then!solve!for!m. 4)!Find!the!C.F!by!using!complimentary!function!rules 5)!Find!the!P.I!by!using!method!of!under determined!coefficients (ONLY!IF!the! auxiliary!equation!≠!0. 6)!Find!x(t) = cC.F. (t) + xP.I. (t) (Or y if you have eliminated x) 7) Substitute x(t) back into the original equation to find y(t) (or y(t) to find x(t) if x was eliminiated) 8) Find the determinant of the two original equation and determine the power of the determinant ( What is the greatest power of D) 9) Check this power is equal to the number of arbitrary constants in the solution x(t) and y(t) 10) If 9) is the same, then the answer is good. If det no of arbitrary constants, then sub back x(t) and y(t) into original solution, work out, and now det should = no of arbitrary constants