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# ch04 - The generalization of the Wronskian is given on page...

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Chapter Review Sheets for Elementary Differential Equations and Boundary Value Problems, 8e Chapter 4: Higher Order Linear Equations Definitions: n th Order Linear ODE Fundamental Set of Solutions, General Solution Homogeneous and nonhomogeneous equation Linear Dependence and Independence Characteristic Polynomial, Characteristic Equation Variation of parameters Theorems: Theorem 4.1.1: Existence and uniqueness of solutions to higher order linear ODE's. Theorem 4.1.2: General solutions to higher order linear ODE's and the fundamental set of solutions Important Skills: The methods for solving higher order linear differential equations are extremely similar to those in the last Chapter. There is simply n times the fun! The general solution to an n th order homogeneous linear differential equation is obtained by linearly combining n linearly independent solutions. (Equation 5, p. 220)
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Unformatted text preview: The generalization of the Wronskian is given on page 221. It is used as in the last Chapter to show the linear independence of functions, and in particular homogeneous solutions. • For the situation where there are constant coefficients, you should be able to derive the characteristic polynomial, and the characteristic equation, in this case each of n th order. Depending upon the types of roots you get to this equation, you will have solution sets containing function similar to those in the second order case. (Examples 2-4, p. 227-229) • The general solution of the nonhomogeneous problem easily extends to the n th order case. (Equation 9, p. 225) • Both variation of parameters, and the method of undetermined coefficients generalize to determine particular solutions in the higher dimensional situation. (Example 3, p. 234; Example 1, p. 239) Relevant Applications: • Double and multiple spring mass systems...
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