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Bruce E. Shapiro, Ph.D.
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Math 351
Lecture Notes on Differential Equations

California State University, Northridge
Spring 2005

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  1. Introduction to Differential Equations (4.5 MB) Revised 6 Feb 2005
    • Ways of Writing Differential Equations
    • One-parameter Families of Solutions
    • Initial Value Problems
    • Concepts of Existence and Uniqueness
    • Differential Equations in the Sciences

  2. Uniqueness of Solutions (3.1 MB) Revised 26 Feb 2005 (the material on direction fields has been moved to chapter 3)
    • General Theory
    • The Lipshitz Condition
    • The Gronwall Inequality

  3. Slopes and Flows (2.7 MB) Revised 16 Feb 2005
    • The Slope (Direction) Field
    • The Flow Field
    • Sketching of Solutions

  4. First Order Differential Equations (4.9 MB) Revised 26 Feb 2005
    • Separable Equations
    • Homogeneous Equations
    • Exact Equations
    • Integrating Factors

  5. First Order Linear Equations (1.7 MB)
    • The Linear Differential Operator
    • Bernoulli Equations

  6. Method of Successive Approximations (3.5 MB)Revised 3 March 2005
    • Picard Iteration
    • Picard-Lindlöff Existence Theorem
    • Continuous Dependence on a Parameter
    • Existence Theorem for Systems

  7. Approximate and Numerical Solutions (3.8 MB)Revised 24 April 2005
    • Euler's Method
    • Epsilon-approximate Solutions
    • Cauchy-Euler Existence Theorem
    • Approximation Error
    • Improved Euler's Method
    • Taylor Series Methods
    • Runge-Kutta Method

  8. Second Order Linear Equations with Constant Coefficients (6.2 MB)Revised 3o April 2005
    • General Theory
    • Factorization of the Differential Operator
    • Characteristic Equation and General Integral Forumula for Solution of Homogeneous Equation
    • General Integral Formula for Particular Solution
    • Method of Undetermined Coefficients
    • Harmonic Oscillations

  9. Higher Order Linear Equations with Constant Coefficients (3.6 MB)
    • Generalization of theory and methods for second order equations to higher order equations

  10. Linear Equations with Variable Coefficients (5.6 MB)
    • Fundamental Matrices and the Wronskian
    • Method of Reduction of Order
    • Method of Variation of Parameters

  11. Series Solutions at Ordinary Points (4 MB)

  12. Series Solutions at Regular Singularities (3.8 MB)

  13. Linear Systems (6.3 MB)Revised 4 May 2005
    • Fundamental Matrices and the Wronskian
    • Exponential of a Matrix
    • Systems with Constant Coefficients
    • Variation of Parameters for Systems

  14. Critical Points of Autonomous Linear Systems (3.5 MB)Revised 17 April 2005

  15. Flows in the Plane --- sorry - not done in time
    • Just read Strogatz Chapters 6 & 7
    • ... Critical points ... Linearization ... Limit Cycles ...
  1. Summary of Methods and Results (2.3 MB)Posted 5 May 2005
    • Summary of basic formulas that may help organize your thoughts for the final ...
  1. Integral Tables




















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