Introduction to Differential Equations

by M. Taylor. Published by the American Mathematical Society

This work began as what is now Chapter 2. The intention was to use this material to supplement Differential Equations texts, which tended not to have sufficient material on linear algebra.

After a while it started to sink in that these texts I intended to supplement had other serious shortcomings. For a fuller discussion of this, see Howl for ODE.

My goal shifted from supplementing these texts to replacing them. Here is a description of the book that resulted.

Chapter 1 treats single differential equations, linear and nonlinear, with emphasis on first and second order equations. The first section provides a self contained development of exponential functions eat, as solutions of the differential equation dx/dt=ax. We allow a to be complex, and also provide a self contained treatment of the trigonometric functions. The following two sections treat first order equations, and then we quickly move to second order equations. We emphasize equations arising from Newton’s laws, applied to 1D motion, first without friction and then with friction. We are motivated to study linearizations of these equations, which occupy the rest of the chapter.

Chapter 2 is devoted to linear algebra. This includes definitions of vector spaces and linear transformations, the notion of basis and dimension of a vector space, and representation of a linear transformation by a matrix, in terms of a choice of basis. We have a treatment of determinants of square matrices, followed by a discussion of eigenvalues and eigenvectors of a linear transformation, and then of generalized eigenspaces. We also discuss several special classes of linear transformations, particularly nilpotent transformations, and also self-adjoint, skew-adjoint, unitary, and orthogonal transformations.

In Chapter 3 we apply the material of Chapter 2 to the study of linear systems of differential equations. The first section is devoted to the matrix exponential, extending the results that began Chapter 1. Section 2 provides a complementary approach to trigonometric functions, and extends the scope to the sine and cosine of matrices. We proceed to various topics on linear systems, first constant coefficient and homogeneous, then non-homogeneous, then variable coefficient. An integral formula, Duhamel’s formula, is seen to provide an elegant replacement for the method of variation of parameters. Several sections are devoted to applications to electrical circuits, spring systems, and the Frenet-Serret equations for space curves. We end the chapter with a treatment of power series methods for systems with analytic coefficients and a treatment of systems with regular singular points, and show how this material applies to single equations of second order.

Chapter 4 treats nonlinear systems of differential equations. We prove results on existence and uniqueness of solutions, and dependence on initial conditions and other parameters. We discuss geometrical aspects, interpreting solving the initial value problem as constructing the flow generated by a vector field. We bring in the phase portrait and discuss the nature of critical points of a vector field. We resume the discussion from Chapter 1 of equations arising from applying Newton’s laws, this time to the interaction of several bodies in multi-dimensional space. We produce Newton’s solution to the planetary motion problem. We discuss variational problems, and show how they allow another attack on problems in Newtonian physics. We also discuss some nonlinear systems that arise in mathematical biology, such as predator-prey equations and competing species equations. We end with a discussion of how, in dimension 3 and higher, flows can have chaotic behavior.

Contents

  • Chapter 1. Single differential equations
    1. The exponential and trigonometric functions
    2. First order linear equations
    3. Separable equations
    4. Second order equations; reducible cases
    5. Newton’s equations for motion in 1D
    6. The pendulum
    7. Motion with resistance
    8. Linearization
    9. Second order constant-coefficient linear equations — homogeneous case
    10. Nonhomogeneous equations I – undetermined coefficients
    11. Forced pendulum — resonance
    12. Spring motion
    13. RLC circuits
    14. Nonhomogeneous equations II – variation of parameters
    15. Variable coefficient second order equations
    16. Bessel’s equation
    17. Higher order linear equations
    18. The Laplace transform
  • Appendices to Chapter 1
    1. The genesis of Bessel’s equation: PDE in polar coordinates
    2. Euler’s gamma function
    3. Differentiating power series
  • Chapter 2. Linear algebra
    1. Vector spaces
    2. Linear transformations and matrices
    3. Basis and dimension
    4. Matrix representation of linear transformations
    5. Determinants and invertibility
    6. Eigenvalues and eigenvectors
    7. Generalized eigenvectors and the minimal polynomial
    8. Triangular matrices
    9. Inner products and norms
    10. Norm, trace, and adjoint of a linear transformation
    11. Self-adjoint and skew-adjoint transformations
    12. Unitary and orthogonal transformations
  • Appendices to Chapter 2
    1. The Jordan canonical form
    2. Schur’s upper triangular form
    3. The fundamental theorem of algebra
  • Chapter 3. Linear systems of differential equations
    1. The matrix exponential
    2. Exponentials and trigonometric functions
    3. First-order systems derived from higher-order ODE
    4. Non-homogeneous equations and Duhamel’s formula
    5. Simple electrical circuits
    6. Second order systems
    7. Curves in R3 and the Frenet-Serret equations
    8. Variable coefficient systems
    9. Variation of parameters and Duhamel’s formula
    10. Power series expansions
    11. Regular singular points
  • Appendices to Chapter 3
    1. Logarithms of matrices
    2. The matrix Laplace transform
    3. Complex analytic functions
  • Chapter 4. Nonlinear systems of differential equations
    1. Existence and uniqueness of solutions
    2. Dependence of solutions on initial conditions and other parameters
    3. Vector fields, orbits, and flows
    4. Gradient vector fields
    5. Newtonian equations
    6. Central force problems and two-body planetary motion
    7. Variational problems and the stationary action principle
    8. The brachistochrone problem
    9. The double pendulum
    10. Momentum-quadratic Hamiltonian systems
    11. Numerical study — difference schemes
    12. Limit sets and periodic orbits
    13. Predator-prey equations
    14. Competing species equations
    15. Chaos in multidimensional systems
  • Appendices to Chapter 4
    1. The derivative in several variables
    2. Convergence, compactness, and continuity
    3. Critical points that are saddles
    4. Blown up phase portrait at a critical point
    5. Periodic solutions of x”+x = e f(x)
    6. A dram of potential theory
    7. Brouwer’s fixed-point theorem
    8. Geodesic equations on surfaces
    9. Rigid body motion in Rn and geodesics on SO(n)