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Differential Equations

Since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations.

Steven Strogatz

Differential equations are equations that involve an unknown function and its derivatives. They are used to model a wide variety of phenomena in the natural and social sciences. We will introduce the basic concepts of differential equations and explore some simple examples.

Previously, in Calculus 1 and 2, we introduced differential equations. Here, we go deeper (hint: DEs go very deep).

Calculus 3 (multivariable calculus) is not a prerequisite for this course, but rather a corequisite.

Wide-Ranging Applications​

Below are some examples of differential equations that are used to model real-world phenomena. It shows how important differential equations are in the sciences.

  1. A simple problem from calculus:

    Problems like this can just be solved by integrating both sides.

  2. Newton's Law of Cooling:

    where is the temperature of an object, is the ambient temperature, and is a constant.

    This is covered in the Calculus 1 and 2 notes.

  3. Newton's Law of Gravitation:

    where is the distance between two objects, is the mass of one of the objects, and is the gravitational constant.

    This is a second order differential equation, but we can convert it to a first order differential equation by introducing a new variable.

  4. Simple harmonic ocsillator; Hooke's Law spring:

    where is the mass of the object, is the spring constant, and is the displacement from equilibrium.

  5. Damped harmonic oscillator:

    where is the damping constant.

  6. Simple pendulum:

    where is the angle of the pendulum, is the acceleration due to gravity, and is the length of the pendulum.

    This was left unsolved in the Calculus 1 and 2 notes. We will solve it in this course.

  7. Heat equation:

    where is the temperature of a substance, is time, and is a proportionality constant known as the thermal diffusivity.

    The symbol is the Laplacian of , which is the sum of the second partial derivatives of with respect to each spatial variable.

Resources​

  • Khan Academy: Differential Equations - Taught by Sal Khan. A good introduction to differential equations. Not much content, though.
  • MIT OCW: Differential Equations - Taught by Arthur Mattuck. A more advanced course on differential equations.
  • Any mechanics book - mechanics is a field that heavily uses differential equations. If you want to see how DEs are used in practice, and get a ton of practice, this is the way to go. Generally, for practice, I like to do mechanics problems instead of DE problems.
  • 3Blue1Brown: Differential Equations - A series of videos by Grant Sanderson. These videos are more about the intuition behind differential equations, rather than the math itself.