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Differential equations are a mathematical tool for describing change, and can be used to model physical systems. In this video, a pendulum is used as an example to show how differential equations can be used to understand the behavior of a system. The equations are complex, but provide insight into how the system behaves. Differential equations can also be used to explore how systems interact, and to simulate the behavior of systems.

**00:00:00**Differential equations describe change in a system, and can be used in physics and other fields. In this tour, we start with simple examples and work our way deeper.**00:05:00**Differential equations are difficult to solve, but can provide a deep understanding of how physical systems behave. In this example, a pendulum is described, and the differential equation describing its motion is shown. The equation is complex, and requires the use of derivatives and higher-order derivatives. The pendulum also experiences air resistance, which is modeled using a derivative. The final equation is complicated, but does provide an approximate solution for the angle the pendulum makes with the vertical.**00:10:00**Differential equations are used to describe the motion of physical systems, such as pendulums.visualizing the state space of a differential equation using 2-dimensional graphs is a useful way to understand the equation and its dynamics.**00:15:00**Differential equations are used to model the behavior of systems, and in particular, the behavior of a pendulum. The equations describe the trajectory of a point in a space filled with trajectories corresponding to different initial conditions. In practice, the space is often reduced to a single dimension by taking advantage of symmetries in the system.**00:20:00**Differential equation models can be used to explore how systems interact, and can be used to simulate the behavior of systems. The video introduces the concept of stability, and explains how it can be determined using a phase diagram. It then demonstrates how to compute theta(t), a measure of position in a differential equation model, using a Python program.**00:25:00**Differential equations are a tool for solving problems, and can be studied in a number of ways. One of these ways is to find exact solutions, which can also help us study more general cases of non-solution. However, even with exact solutions, there are limits to how well we can predict outcomes.

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