This 12-minute video lesson shows how to figure out the Laplace Transform of the Dirac Delta Function. [Differential Equations playlist: Lesson 42 of 45]

This 24-minute video lesson provides an introduction to the unit step function and its Laplace Transform. [Differential Equations playlist: Lesson 38 of 45]

This 10-minute video lesson shows the Laplace Transform of t^n: L{t^n}. [Differential Equations playlist: Lesson 37 of 45]

This 11-minute video lesson shows how to use the Laplace Transform to solve an equation we already knew how to solve. [Differential Equations playlist: Lesson 32 of 45]

This 11-minute video lesson provides a grab bag of things to know about the Laplace Transform. [Differential Equations playlist: Lesson 34 of 45]

This 9-minute video lesson gives an example of using initial conditions to solve a repeated-roots differential equation .[Differential Equations playlist: Lesson 21 of 45]

This 12-minute video lesson looks at what happens when the characteristic equation only has one repeated root. [Differential Equations playlist: Lesson 20 of 45]

This 12-minute video lesson introduces separable differential equations. [Differential Equations playlist: Lesson 2 of 45]

This 14-minute video lesson looks at 3 basic differential equations that can be solved by taking the antiderivatives of both sides.

This 14-minute video lesson explains how the product of the transforms of two functions relates to their convolution. [Differential Equations playlist: Lesson 44 of 45]

This 10-minute video lesson looks at using the method of undetermined coefficients to solve nonhomogeneous linear differential equations. [Differential Equations playlist: Lesson 22 of 45]

This 11-minute video lesson gives another example using undetermined coefficients. [Differential Equations playlist: Lesson 23 of 45]

This 8-minute video lesson gives another example where the nonhomogeneous part is a polynomial. [Differential Equations playlist: Lesson 24 of 45]

This 6-minute video lesson concludes the series on undetermined coefficients by putting it all together. [Differential Equations playlist: Lesson 25 of 45]

This 12-minute video lesson shows how to use the convolution Theorem to solve an initial value problem. [Differential Equations playlist: Lesson 45 of 45]

This 19-minute video lesson shows how to solve a non-homogeneous differential equation using the Laplace Transform. [Differential Equations playlist: Lesson 35 of 45]

Watch this video to learn about the working of the differential in tractors and the purpose of final drive in tractor.

In this video, we use direction fields (drawn as quiver plots) to illustrate the numerical integration of differential equations. We include a heuristic example of how one might try to adapt step size by comparing different orders of approximation.

In the first video, we present a biological circuit topology from Ma, Trusina, El-Samad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760-773 (2009). This topology supports adaptation, which is not the absence of change in response to stimulation/stress, but, instead, the ability to produce delayed compensation for those changes. In the second video, we summarize the method of almost linear stability analysis used to solve for the dynamics of this example system.

To describe how oscillations are supported in systems of differential equations, we present a classic "Romeo and Juliet" picture of two-dimensional oscillations, and we analyze how trajectories change as nullclines are arranged at different angles in the phase plane. In addition to models based on traditional systems of differential equations, dynamical systems with time delays and dynamical systems with stochastic fluctuation (i.e. stochastic resonance) can also support oscillations.