Section

Content

Suggested Problems

Section 3.1

 InitialValue and Boundary Value Problems
 Existence and Uniqueness of IVP
 Potential NonExistence/NonUniqueness of BVP
 Homogeneous Equations
 Operator Notation
 Vector Space of Solutions of Homogeneous Equation
 Linear Independence of Solutions of Homogeneous Equation
 Theorem 3.3 Wronskian Test for Linear Independence
 Fundamental Set of Solutions
 Superpostion Principle  NonHomongenous Equations


Section 3.2

 Reduction of Order
 Standard Form
 y_{2} = u(x)y_{1}(x)


2, 6, 9 ,11

Section 3.3

 Linear, Constant Coefficient, Homogeneous
 Auxillary or Characteristic Equation
 Case I: Distinct Real Roots
 Case II: Repeated Real Roots
 Case III: Complex Conjuate Roots
 Special Cases
 y" + k^{2}y = 0
 y"  k^{2}y = 0

3, 5, 10, 13, 15, 18, 24, 31, 33, 34

Section 3.4

 Method of Undetermined Coefficients
 Particular Solution L(y) = f(x)
 f(x) = polynomial
 f(x) = exponential
 f(x) = cosine or sine
 Glitch in the Method
 InitialValue Problem

4, 8, 12, 16, 19, 23, 28, 31

Section 3.5

 Variation of Parameters
 Standard Form
 Construction Fundamental Set of Solutions for Homogeneous Problem
 Solution for u_{1}' and u_{2}'


1, 2, 6, 8, 11

Section 3.6

 CauchyEuler Equation
 Auxillary or Characteristic Equation
 Case I: Distinct Real Roots
 Case II: Repeated Real Roots
 Case III: Complex Conjuate Roots
 NonHomogeneous Equations

1, 4, 9, 11, 15, 19

Section 3.8

 Linear Dynamical Systems
 Hooke's Law
 Newton's Second Law
 Spring/Mass
 Equilibrium Position
 Coordinate System Orientation
 Free Undamped Motion
 mx" + kx = 0
 x" + ω^{2}x = 0
 Alternative Form of Solution
 Free Damped Motion
 mx" + βx' + kx = 0
 x" + 2λx' + ω^{2}x = 0
 Overdamped, Critically Damped, Underdamped
 Driven or Forced Motion
 mx" + βx' + kx = f(t)
 x" + 2λx' + ω^{2}x = F(t)
 Transient vs SteadyState Solutions

3, 6, 9, 10, 21, 22, 26, 29
