MATH 280, Study Guide for the Final Exam

8/18/2014

Key concepts for Chapter 7 (Refer to the guides to Midterm 1 and Midterm 2 for the rest of the topics. Find them in the textbook/lecture notes, know for the test.)

• Laplace transform: definition as an integral
• Properties: linearity, scaling, derivatives: multiplication by t or s; translations, unit step function: multiplication by e^t or e^-s; inverse Laplace transform
• Method of solving initial-value problems with Laplace transform; partial fractions
• Differential equations with piecewise functions; representation of piecewise functions using the unit step function
• The Dirac delta function, its Laplace transform
• Boundary value problems for the beam equation with different types of boundary conditions: embedded end, simply supported, free end; method of solution with Laplace transform

Key types of exam questions

• Verify that a function is a solution to a differential equation/initial value problem; determine interval of existence (1.1: 19, 20, 25, 43)
• Solution curves without a solution: phase line, stability of equilibrium solutions (2.1: 21-28, 31, 38, 39; 1.1: 58)
• Check conditions for the existence/uniqueness theorem; give examples of non-unique solutions (1.2: 15-24, 31)
• Modeling with first-order equations: population models; mixing problems; Newton's law of cooling (3.1: 5-7, 13-15, 25 (derive the differential equations in each case); 3.2: 2, 3; 2.1: 38, 39)
• Given a first-order equation, choose an appropriate method to solve it; find the general solution (see homework problems for Chapter 2)
• Set up equations to solve an ODE by Euler's method; compute a few steps of approximate solution (2.6: 1, 3, 4)
• Fundamental set; Wronskian, linear independence (4.1: 28, 29, 39)
• Solve a linear equation of order 2 or higher; find a particular solution by undetermined coefficients or variation of parameters (see homework problems for Chapter 4)
• Special cases of second-order equations: Cauchy-Euler, Nonlinear equations solved by a substitution u=y' (see homework problems for Chapter 4)
• Modeling with second-order equations: mass/spring systems, beam models (5.Rev: 12, 15; 5.2: 5; 7.4: 77; 7.5: 13, 14)
• Eigenvalue problems with different kinds of boundary conditions; find eigenvalues and eigenfunctions (5.2: 9-15, 22)
• Power series method: ordinary/singular points. Find recurrence relation; compute a few terms of the power series solution; estimate the radius of convergence (6.Rev: 3, 4, 9-11)
• Verify (prove) a property of the Laplace transform; compute a Laplace transform using the properties (7.1: 50, 55; Theorems 7.2.2, 7.3.1-2 / lecture notes)
• Solve a linear equation with Laplace transform (see homework problems for Chapter 7)
• Examples with delta functions; equations of beams (see homework problems for Chapter 7)

Good luck on the exam!