Math 240 - Differential Equations

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  Syllabus*         Spring 2000  
  Midterm I*     Spring 1999     Spring 2000  
  Midterm II*     Spring 1999     Spring 2000  
  Final*     Spring 1999     Spring 2000  
  • Detailed weekly schedule
  • Topics causing problems
  • Exam rules and terms
  • Class roster
  • *The files are in .pdf format. Download Acrobat Reader here. Postcript or .dvi files are available upon request no spam
    Week  Topic  Notes
    1    Feb.7-11    Introduction to Differential Equations   
    2    Feb.14-18    First Order O.D.E. Separation of Variables   
    3    Feb.21-25    Integrating Factors. Homogeneous Equations   
    4    Feb.28-Mar.3    Applications of First Order O.D.E.     (**) 
    5    Mar.6-10    Linear Dependence. Wronskian     (**) 
    6    Mar.13-15    Reduction of Order   
    7    Mar.20-24    Method of Undetermined Coefficients    Midterm I  
    8    Mar.27-31    Variation of Parameters   
    9    Apr.3-7    Applications of Second Order O.D.E.   
    10    Apr.10-14    Inverse Differential Operators*   
    11    Apr.17-21    Singularities of Differential Equations*   
    12    Apr.24-28    Series Solutions    Midterm II  
    13    May 1-5    Series Solutions   
    14    May 8-12    The Laplace Transform   
    15    May 15-18    The Laplace Transform   
      Finals week   Final  

    * These topics are likely to be skipped

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    Midterm I     (25%)   March 24, 2000, 1:40 pm     See   important remarks

    Topics covered (tentative):
  • The notion of differential equation; classes of equations
  • Solutions: definition, existence, uniqueness
  • First order equations: separation of variables, homogeneous equations, linear equations
  • Integrating factors
  • Special equations/methods: substitution, linear coefficients, Bernulli's equation
  • Applications of first order equations     (see problematic topics)

    Chapters:     1, 2, 4, 5(1-5).
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    Midterm II     (25%)   April 28, 2000, 1:40 pm     See   important remarks

    Topics covered (tentative):    
  • Higher order linear equations: existence and uniqueness of solutions; independence of solutions, Wronskian; structure of the general solution to a homogeneous/nonhomogeneous equation; operator notation
  • Homogeneous linear equations with constant coefficients
  • Nonhomogeneous equations: method of undetermined coefficients
  • Nonhomogeneous equations: variation of parameters
  • Reduction of order
  • Applications of second order linear O.D.E. (pay special attention to harmonic oscillators, such as spring, oscillating circuit, simple pendulum). You can use formulas for damped and undamped vibration; however, in certain cases (say, forced vibration) considering the equation might still be necessary (see, e.g., Problem 1 in 1999 Final)
  • Nonlinear equations: reduction of order (see 16.8, 16.9)

    Chapters:     6, 7, 8, 9, 10, 16(8, 9).
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    Final     (40%)   See your class schedule     See   important remarks

    Topics covered (tentative):    
  • Power series solution to a linear differential equation at an ordinary point. Radius of convergence
  • Power series solution at a regular singular point. The indicial equation
  • The Laplace transform: definition, basic properties, transforms of particular functions
  • The Gamma function
  • The inverse transform: basic properties, techniques, table of transforms
  • Application of the Laplace transform to solving linear O.D.E.'s (especially, with discontinuous right hand side part)
  • The convolution theorem

    Chapters:     14, 15(1-5), 17, 18(1-9, 12). (Of course, the material covered by the midterms is also included.)
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    Quizzes     (10%)   Held weekly on Tuesday in class (10-15 mins at the end) except midterm weeks. Hopefully 10 quizzes will be given, with grading out of 8 best.

    All questions regarding the quizzes are to be directed to the assistant. The assistant is instructed to  give no credit  to identical papers. The same applies to the exams.
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    During the exams please keep in mind the following:
  • Calculators are  not  allowed
  • Identical solutions (especially identically wrong ones) will  not  get credit. I reserve the right to decide what "identical" means. You still have the right to complain
  • Do not argue about the distribution of the credits among different parts of a problem. I only accept complaints concerning my misunderstanding/misreadung your solution
  • Show all your work. Correct answers without sufficient explanation might  not  get full credit
  • Indicate clearly and unambiguously your final result. In proofs, state explicitly each claim
  • Do not misread the questions or skip parts thereof. If you did, do not complain
  • If you believe that a problem is misstated, do not try to solve it; explain your point of view instead
  • Each problem has a reasonably short solution. If your calculation goes completely out of hands, something must be wrong

    Grading policy
    I will take off a few (2-3) points for arithmetical mistakes. However,  a lot  of points will be taken off for `obvious' mistakes, i.e., either those that you can easily avoid or those showing a deep misunderstanding of the subject. This includes, but is not limited to, the following:
  • Wrong dimension in a physical problem
  • Things that don't make sense (e.g., an equation like x+ y+ x dx+ y dy = 0, not homogeneous in dx, dy)
  • `Too straightforward' integration (e.g., integrating the equation dv = v dx to v = vx+ C)
  • When solving an exact equation, appearing of a wrong variable in the last equation to determine the `constant' C(y) (e.g., C'(y) = x+ y)
  • Wrong number of constants in a general solution
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