MAT 275 – Spring 2009

Modern Differential Equations

• Review all problems on homework, WeBWorK, and labs. Be able to work any of these problems easily without looking at your notes or the book. Several exam items will be modifications of these problems.
• Know the basic terminology from the course: ordinary differential equation, solution to an ODE, general solution, particular solution, initial value problem, initial conditions, order of an ODE, slope field, autonomous ODE, equilibrium solution, stable, unstable, semi-stable, linear ODE, local error, cumulative error, roundoff error, order of a numerical method, adaptive step size.
  
• Understand and be able to interpret and translate between various representations for an ODE
• Algebraic form dy/dx = f (x,y).
• Slope field.
• Graph of a solution or a family of solutions y vs. x.
  
• Graph of dy/dx vs. y for an autonomous ODE.
• Output from a numerical method.
  
• Understand geometrically why the condition that f_y(x,y) be continuous is necessary to have a unique solution to an ODE dy/dx = f (x,y).
• Be able to derive the ODE models we have covered in class or on homework: Torricelli's Law, mixture problems as in §1.5, population growth (justifying why rate dP/dt should be proportional to amount P and discussing the idea of "birth rate"), Newton's law of cooling (justified in terms of its reasonableness rather than deriving it), the path followed by an object being pulled on a string (from Lab 2).
  
• Carefully review all of the material in the handout on Error Analyses:
• Know the basic idea behind the Euler, improved Euler, and Runge-Kutta methods.
• Know the meaning of all elements of a numerical method and their algebraic and graphical representations: initial condition, step size, approximation, actual value, error, local error, cumulative error, slopes from the ODE at various points.
  
• Be able to justify the formula for Euler's method based on using a tangent line to generate an approximation.
• Understand the two different sources of error in the Euler method. Understand that each of these are repeated for each step of the method so that they compound (rather than simply add).
  
• Understand geometrically how y"(x) and f_y(x,y) are the relevant quantities that determine how large the error from the two sources can be.
• Be able to derive the bound ½Mh² for the local error (found at the top of p. 2).
• Be able to derive the recursive relationship for the cumulative error at each step (found in the middle of p. 2).
• Understand the meaning of the methods being first-order, second order, and fourth-order and know what this means in terms of the relationship between step size and cumulative error (alternatively the relationship between number of steps and cumulative error).
• Be able to derive the condition for choosing the adaptive step size for the Runge-Kutta method (found on pp. 4-5). You may assume that the compounding introduces the factor He^LH.
  
• Understand what roundoff error is, its relationship to step size, and the tension between roundoff error and cumulative (truncation) error for small step sizes.

• Review all problems on homework, WeBWorK, and labs. Be able to work any of these problems easily without looking at your notes or the book. Several exam items will be modifications of these problems.
• Know the basic terminology from the course (in addition to important terminology from Exam 1 content): linear second order ODE, homogeneous, nonhomogeneous, constant coefficients, characteristic equation, associated homogeneous equation, complementary solution, particular solution, general solution, amplitude, angular frequency, frequency, period, phase angle, time lag, damped, undamped, overdamped, underdamped, critically damped, forced, free, beat, beat frequency, resonance, angular velocity, transient solution, steady periodic solution, Laplace transform, linear operator, inverse operator.
  
• Know the principle of superposition, when it is applicable (and not), and how it relates to determining general solutions.
• Know how to suppose a solution of the form e^rt for a second order constant coefficient linear ODE to generate the characteristic equation, and determine the general solution. Know how to handle cases for distinct roots, complex roots, and repeated roots. Be able to determine what type of behavior results from various combinations of the constants.
• Be able to explain why the general solution of a nonhomogeneous linear ODE is the sum of the general solution to the associated homogeneous equation plus any particular solution to the nonhomogeneous equation. Know how to implement this to find the general solution to a nonhomogeneous equation.
• Be able to use the method of undetermined coefficients to find a particular solution to a nonhomogenous ODE. Know how to handle cases where the external force is a solution to the associated homogeneous equation.
  
• Be able to construct ODE models: spring-mass-dashpot systems using Newton's second law and Hooke's law, RLC circuits using Kirchoff's law (potential drops for components will be provided), mechanical systems involving linear and rotational motion using conservation of energy (you should know the kinetic energy and potential energy formulas, but not formulas for moment of inertia)
  
• Be able to determine the amplitude, angular frequency, frequency, period, phase angle, and time lag for solutions. Also be able to identify these properties graphically. You'll need to be able to rewrite an expression of the form Acos(ωt)+Bsin(ωt) as Ccos(ωt-α).
  
• Know how resonance arises (contextually and algebraically), and how to characterize such solutions.
• Know the definition of the Laplace transform.
• Be able to derive the Laplace transforms of the following functions: 1, t^a, tⁿ, e^at, and u_c(t)=u(t-c). (You'll need to know the definition of the Gamma function to derive the Laplace transform of t^a.)
• Be able to derive the formulas for the Laplace transforms of the first and second derivatives of a function.
• Be able to use the table to determine the Laplace transforms and inverse Laplace transforms of various functions.
• Be able to explain the idea behind using Laplace transforms to solve IVPs. Be able to explain why it works and the key properties of the Laplace transform that make this technique possible.
• Be able to solve IVP's using Laplace transforms.
• Be able to work with step functions, delta functions, and the greatest integer function (e.g., writing new functions involving these) and their Laplace transforms.
• Be fluent with partial fractions.
• Be prepared for a problem very similar to one from Exam 1.
  

• Review all problems on homework, WeBWorK, and labs. Be able to work any of these problems easily without looking at your notes or the book. Several exam items will be modifications of these problems.
• Know the basic terminology from the course (in addition to important terminology from Exam 1 and Exam 2 content): system of ordinary differential equations, solution to a system of ODEs, order of a system, solution curve or trajectory, direction field for a system, linear system, matrix, elements, rows and columns of a matrix, identity matrix, matrix multiplication, determinant, eigenvector, eigenvalue, straight-line solution, characteristic equation, defective eigenvalue, generalized eigenvector,
  
• Be able to read and express a system of ODEs and initial conditions in any of the following formats: a list of equations (for x'(t)=f(x,y) and y'(t)=g(x,y), an equation for the vector [x';y']=[f(x,y);g(x,y)], or an equation using a matrix [x';y']=A[x;y].
• Be able to find the eigenvectors and eigenvalues for a two-dimensional linear system using either the eigenvector-first approach or the eigenvalue-first approach.
  
• Be able to interpret and express the meanings of eigenvectors and eigenvalues in terms of the linear system of ODEs
• Know what straight-line solutions are and understand the conditions on their slope (y'/x'=y/x) and eigenvalue ([x';y']=λ[x;y]). Be able to relate these ideas to the eigenvectors and eigenvalues.
  
• Know and be able to derive the condition that a linear system of ODEs has infinitely many equilibrium points. Be able to explain this geometrically. Be able to relate this to λ=0 being an eigenvalue.
  
• Know and be able to derive the condition for any number λ to be an eigenvalue.
• Be able to determine the general solution to a system of equations with distinct real eigenvalues, complex eigenvalues, one eigenvalue with two independent eigenvectors, and one eigenvalue without a second independent eigenvector.
  
• Be able to read and express solutions in any of the following formats, a list of equations x(t)=... and y(t)=..., an equation for the vector [x;y]=[x(t);y(t)], or an equation using the eigenvectors, e.g., [x;y]=c₁v₁e^λ1t+c₂v₂e^λ2t.
• Be able to draw and interpret phase plane diagrams for solutions to  linear systems.
  
• Be able to convert a second-order ODE into a system of first order ODEs, solve, and interpret the solutions and phase plane diagrams.