MATH 280, Study Guide for Midterm Test 2

8/12/2014

Test coverage

1. Higher-Order Differential Equations (Chapter 4)
   1. Linear equations: Basic Concepts (homogeneous/nonhomogeneous equations, linear superposition principle; existence/uniqueness; Wronskian) (4.1)
   2. Reduction of Order (4.2)
   3. Homogeneous Equations with Constant Coefficients (4.3)
   4. Non-Homogeneous Equations: Method of Undetermined Coefficients (4.4)
   5. Variation of Parameters (4.6)
   6. Cauchy-Euler Equation (4.7)
   7. Nonlinear Equations of Higher Order (4.10)
2. Modeling with Higher-Order Differential Equations (Chapter 5)
   1. Linear Models: Initial-Value Problems (5.1)
   2. Linear Models: Boundary-Value Problems (5.2)
3. Series Solutions of Differential Equations (Chapter 6)
   1. Review of Power Series (6.1)
   2. Solutions about Ordinary Points (6.2)
   3. Solutions about Regular Singular Points: Method of Frobenius (6.3)

Key concepts (find in the textbook/lecture notes -- know for the test)

• Linear higher-order equations: Existence/Uniqueness Theorem
• Homogeneous and non-homogeneous linear equations; linear superposition principle
• Linearly independent solutions; the Wronskian, fundamental set of solutions
• Method of reduction of order and the formula for the second solution
• Equations with constant coefficients; reduction to a quadratic/polynomial auxiliary equation
• 3 Different cases of homogeneous solutions, depending on the roots of the auxiliary equation
• Method of undetermined coefficients (superposition approach)
• Determinants for 2X2 and 3X3 matrices; determinant expansion along a row or column
• Variation of parameters: the method and the determinant formulas for the solutions
• Cauchy-Euler differential equations; form of auxiliary equations for the Cauchy-Euler equations
• 3 Different cases of solutions of Cauchy-Euler, depending on the roots of the auxiliary equation
• Nonlinear equations; substitution u=y' applied to 2 cases of nonlinear equations
• Modeling: mass/spring systems, with damping and external forces
• Overdamped, underdamped, critically damped cases, depending on the roors of the auxiliary equation
• Steady state and transient solutions; resonance
• Fourth-oder linear model of beams; different types of boundary conditions
• Eigenvalue problem for a second-order equation with different types of boundary conditions: eigenvalues and eigenfunctions
• Examples of buckling of a thin vertical column and a rotating string
• Series solutions of differential equations; analytic functions; ordinary and singular points
• Radius of convergence of a power series solution: distance to the closest singular point in the complex plane
• Recurrence relations; obtaining a few first terms of the series, or finding a formula for the coefficients
• Singular points; regular singular points
• Solutions near regular singualr point: the Frobenius method
• Indicial equation; recurrence relations in Frobenius method

Review questions:

4.Rev: 3, 4, 6, 7, 9, 11, 13, 22, 25, 26, 32; 5.Rev: 2, 3, 5, 6, 11, 12, 17, 18; 6.Rev: 3, 4, 7, 8, 15 (find the formula for the coefficients, and write out explicitly the first few terms); 19, 20.

Homework problems contain all the main types of possible exam questions...

Good luck on the exam!