Ordinary Differential Equations

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.

In the course on ordinary differential equations, we usually deal with the following topics:

  • Separable equations

  • First-order linear differential equations

  • Bernoulli differential equations

  • Exact differential equations and test for exactness

  • Homogeneous functions and equations

  • Riccati, Clairaut's, and Lagrange's equations

  • Orthogonal trajectories

  • The general form of a second-order linear differential equation

  • Second-order linear homogeneous differential equations with constant coefficients (real, complex, and repeated roots)

  • Reduction of order

  • Linear independence and Wronskian

  • Nonhomogeneous linear second-order differential equations

  • Undetermined coefficients and variation of parameters

  • Higher-order differential equations

  • Cauchy-Euler differential equations of order n

  • Power series solutions of differential equations

  • Regular singular points

  • The generalized power series method in solving differential equations

  • Definition of integral transforms

  • Definition of Laplace transform

  • A short table of Laplace and inverse Laplace transforms

  • Step functions and their Laplace transforms

  • Computation of Laplace transform of a function's derivative and antiderivative

  • Differentiation and integration of Laplace transforms

  • Dirac delta function

  • Convolution integral

  • Completion of the table of Laplace transforms

  • Systems of differential equations

  • Real, complex, and repeated eigenvalues and their applications in solving systems of differential equations

  • Nonhomogeneous systems of differential equations

For more details, refer to the following book:

Boyce, W. E., DiPrima, R. C., & Meade, D. B. (2017). Elementary differential equations. John Wiley & Sons.

Single-variable Calculus

Multi-variable Calculus

Engineering Mathematics

Linear Algebra and its Applications

Professor Dara Moazzami (mathematician)

For differential equations, a book (in Persian) by Dara Moazzami (pictures) has been also always very useful.