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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

25 hours of lectures.

32 lectures.

Assignments, recitations and exams with solutions(solutions to final exam are not available).



Arthur Mattuck is an Emeritus Professor of Mathematics at MIT. He has been a major force in the design of undergraduate mathematics classes at MIT. Professor Mattuck taught 18.03 many times and his lecture videos and written notes are used throughout this OCW Scholar course.



Haynes Miller is a Professor of Mathematics at MIT. In 2005 he was an MIT MacVicar Faculty Fellow in recognition of his outstanding contributions to undergraduate education. He has taught 18.03 many times and was the prime mover behind its current design. Professor Miller contributed many of the materials used in this OCW Scholar course. 



Creative Commons License
Differential Equations by Prof. Arthur Mattuck, Prof. Haynes Miller, Jeremy Orloff, Dr. John Lewis is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at

Course content

  • The Geometrical View of y'= f(x,y)

  • Euler's Numerical Method for y'=f(x,y)

  • Solving First-order Linear ODEs

  • First-order Substitution Methods

  • First-order Autonomous ODEs

  • Complex Numbers and Complex Exponentials

  • First-order Linear with Constant Coefficients

  • Continuation

  • Solving Second-order Linear ODE's with Constant Coefficients

  • Continuation: Complex Characteristic Roots

  • Theory of General Second-order Linear Homogeneous ODEs

  • Continuation: General Theory for Inhomogeneous ODEs

  • Finding Particular Solutions to Inhomogeneous ODEs

  • Interpretation of the Exceptional Case: Resonance

  • Introduction to Fourier Series

  • Continuation: More General Periods

  • Finding Particular Solutions via Fourier Series

  • Introduction to the Laplace Transform

  • Derivative Formulas

  • Convolution Formula

  • Using Laplace Transform to Solve ODEs with Discontinuous Inputs

  • Use with Impulse Inputs

  • Introduction to First-order Systems of ODEs

  • Homogeneous Linear Systems with Constant Coefficients

  • Continuation: Repeated Real Eigenvalues

  • Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients

  • Matrix Methods for Inhomogeneous Systems

  • Matrix Exponentials

  • Decoupling Linear Systems with Constant Coefficients

  • Non-linear Autonomous Systems

  • Limit Cycles

  • Relation Between Non-linear Systems and First-order ODEs

  • Assignments

  • Recitations

  • Exam(s)

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