Math 4009 Ordinary Differential Equations

Direction Field

MATH 4009 Catalogue Description

Three credits. Three hours of lecture per week. Prerequisite: MATH 3152. Recommended MATH 4031 (Linear Algebra)

Ordinary differential equations of the first order. Linear differential equations with constant coefficients. Linear differential equations of the second order. Systems of differential equations. Applications.


The theory of differential equations plays a fundamental role in almost every branch of science, engineering and economics. By means of differential equations we can describe the evolutionary processes of systems possesing the properties of determinacy and differentiability. The time evolution of the wave function of a quantum mechanical system is described by Schrödinger’s equation. The trajectory of a particle in a gravitational field is described by Newton’s equation. A differential equation describes the form of the adiabats of a homogeneous fluid body. Another describes the flow of heat in such a body. Seismic waves are described by the wave equation. The axioms of electromagnetism are expressed by Maxwell’s equations. Differential equations play an important role in analysis. The exponential function may be defined as the solution to the differential equation x^{\prime} = x with initial condition x(0) = 1. This course will provide and introduction to what are known as ordinary differential for students who have completed a standard calculus sequence and who are conversant with basic linear algebra.

Semester II, 2016-17

Sections 1-8 of the syllabus below will be covered. Depending on the progress of the students, this will be supplemented by a subset of sections 9-12.

General Information


  1. Ordinary Differential Equations – Introduction
  2. Separable ODE’s
  3. Properties of the Direction Field
  4. Integrating Factors
  5. Second Order Linear ODE’s
  6. Second Order Linear ODE’s with Constant Coefficients
  7. Forced Damped Oscillatory Systems
  8. Method of Variation of Parameters
  9. Convolution and Time Invariant Linear Systems
  10. Forced Oscillator – Steady State Solution by Fourier Transform
  11. Scaling of First and Second Order Linear Equations (nondimensionalization)

Old Exams

  1. Sample Exam Questions
  2. Exam 1 – Semester I – 2015
  3. Exam 2 – Semester I – 2015
  4. Exam 3 – Semester I – 2015

Exercise Sets

  1. Geometric Methods
  2. Potential Energy
  3. Separable Differential Equations
  4. Applications
  5. First Order Linear Differential Equations
  6. Autonomous Differential Equations
  7. Differential Equations with Symmetry
  8. Exact Differential Equations
  9. Homogeneous Differential Equations
  10. Second order Linear Differential Equations
  11. Simple Harmonic Motion
  12. Linearity
  13. The Wronskian
  14. Variation of Parameters
  15. Reduction of Order


Your grade will be based upon two partial exams, homework and a comprehensive final exam and is “non negotiable”. Extra credit will not be given. The grading scale will be no worse than the following:   A: 90-100;    B: 80-89;    C: 65-79;    D: 55-64;   F: < 55.   Copying or other forms of cheating will result in an automatic F for the course.   In accord with UPR regulations, persistent lateness or unexcused abscence from class may result in a failing grade and loss of financial support.


  1. Elementary Methods.
    1. The differential equation of first order x^{\prime} = f(t,x) determined by a real valued function of two real variables.
      1. Definition of a solution.
      2. Analytic and Geometric Interpretations
      3. Isoclines and Direction fields.
      4. Lower and upper fences.
    2. The simple linear differential equations of first order x^{\prime}=f(t) determined by a continuous function.
      1. The fundamental theorem of calculus viewed as an existence theorem for the initial value problem x^{\prime} = f(t), \quad x(t_0) = x_0.
      2. Uniqueness of maximal solutions.
    3. Differential equations with separable variables.
    4. Linear differential equations of first order.
      1. Solution by integrating factor.
  2. Review of linear algebra
    1. Definition of a vector space and examples.
    2. The vector spaces C^k[R]
    3. The subspace criterion.
    4. Maximal linearly independent and minimal spanning sets.
    5. Bases and the definition of dimension of a vector space.
    6. Linear maps.
    7. The linear differential operator D: C^1[R] \rightarrow C^0[R] given by f \mapsto f^{\prime} and properties.
    8. Higher order differential operators.
    9. The kernel of a linear map.
    10. Linear equations and vector spaces.
  3. Linear differential equations of first order.
    1. Proof of existence and uniqueness of solutions.
    2. Kernel of L: C^1[R] \rightarrow C^0[R] , where L(f) = p(t)f.
    3. Solution by variation of parameters.
  4. Linear differential equations of second order.
    1. Existence and uniqueness of solutions.
    2. Dimension of the solution space of the homogeneous equation.
    3. The Wronskian and linear independence.
    4. The inhomogeneous second order linear differential equation.
      1. Method of undetermined coefficients.
      2. Variation of Parameters
  5. Linear differential equations with constant coefficients.
    1. Bases for the kernel in the cases where the characteristic polynomial has distinct real roots, repeated roots, complex roots.
    2. Qualitative and quantitative analysis of the forced damped oscillator equation:

          \[mx^{\prime \prime}+ \omega_0^2x^{\prime} + kx= F_0 \cos wt.\]

      1. Cases of zero damping, small and heavy damping.
      2. Resonance and beats. Amplitude modulation.
      3. Solution techniques using complex numbers.
      4. Solution by variation of parameters and derivation of the Greens function.
      5. Convolution and impulse response.
    3. Linear equations of higher order.
  6. Autonomous differential equations and systems
    1. The initial value problem:
                 x^{\prime} = v(x).
                 x(t_0) = x_0
      where v is continuous on an interval.


      1. Implicit form of the solution.
      2. Existence of solutions.
      3. Examples of non uniqueness of solutions when v(x_0) = 0.
      4. Proof of uniqueness at a singular point when v is differentiable.
    2. Symmetry considerations.
      1. Time invariance
      2. The cases when v is odd or even.
    3. The phase space of a differential equations of first order.
      1. Phase line and phase space.
      2. Stability of solutions.
      3. Classification of singular points.
  7. Separable differential equations: y^{\prime} = g(y)/f(x)
    1. The relation between trajectories of the separable autonomous system:
                 x^{\prime} = f(x)
                 y^{\prime} = g(y)
      and solutions of separable differential equations.
    2. Homogeneous differentiable equations.
  8. The differential equation y^{\prime} = v_2(x,y)/v_1(x,y) determined by the vector field (v_1,v_2).
    1. Relation between integral curves of y^{\prime} = v_2(x,y)/v_1(x,y) and trajectories of the first order system
                dx/dt = v_1(x,y)
                dy/dt = v_2(x,y)
    2. Exact differential equations and their relation with conservative fields.
    3. The “closed” necessary condition for exactness; sufficiency on a rectangle or simply connected set.
    4. Integrating factors.
    5. Orthogonal trajectories.
  9. Oscillation Theory (If time permits)
    1. Oscillation theory for linear differential equations of second order.
    2. Self adjoint linear differential equations of second order.
    3. The Sturm separation theorem.
    4. The Sturm comparison theorem.
  10. The Laplace Transform. (If time permits)
    1. Definition of the Laplace transform
    2. Convergence.
    3. Shift and other theorems.
    4. Convolution.
    5. The Laplace transform of a differential equation with non-constant coefficients.
  11. Power Series Solutions (If time permits)
    1. Convergence of series.
    2. Remarks on the nature of functions and power series.
    3. Solution in series of first and second order equations.
    4. Singular points.
    5. Frobenius series and method.
    6. The indicial equation.
    7. The Bessel equation.
  12. Systems of linear differential equations. (If time permits)
    1. Existence theorems.
    2. Fundamental systems of solutions.
    3. Linear systems with constant coefficients.
    4. Matrix notation


  1. V. I. Arnold, Ordinary Differential Equations, MIT Press.
  2. D. K. Arrowsmith and C. M. Place, Ordinary Differential Equations. Chapman and Hall 1982.
  3. Garrett Birkhoff, and Gian-Carlo Rota, Ordinary Differential Equations, New York, NY: John Wiley, 1969, 1989. Fourth Edition.
  4. William E. Boyce, and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, New York, NY: John Wiley, 1969, 1992. Fifth Edition.
  5. Earl A. Coddington, An Introduction to Ordinary Differential Equations, Mineola, NY: Dover, 1989.
  6. Morris W. Hirsch and Stephen Smale, Differential Equations, Dynamical Systems and Linear Algebra, (Academic Press 1975).
  7. J. H. Hubbard and B. H. West5, Differential Equations, A Dynamical Systems Approach, (Springer 1991).
  8. Walter Leighton, A First Course in Ordinary Differential Equations, Wadsworth 1976.
  9. David A. Sanchez, Ordinary Differential Equations and Stability Theory: An Introduction Mineola, NY: Dover, 1979.
  10. George F. Simmons and John S. Robertson, Differential Equations with Applications and Historical Notes, New York, NY: McGraw-Hill,

Electronic Resources

  1. The Geometrical View of y’= f(x,y). MIT Lecture
  2. Wolfram Alpha
  3. DFIELD – Java Program for Direction Fields