This is why you're learning differential equations

Differential equations describe how nature works and are fundamental to understanding the universe. They are used in various fields such as fluid dynamics, electromagnetism (Maxwell's equations), electric circuits, and describing motion, even without physical contact like orbiting bodies.

Just like algebraic equations, differential equations can convey meaning. An example is presented where an equation asks for a curve whose area under the curve is twice its arc length over any given interval. This involves setting up and solving a differential equation derived from the integral definitions of area and arc length.

The video uses a scene from the TV show 'Numb3rs' to illustrate pursuit curves. This involves two objects, one chasing the other, with the chasing object always moving towards the target. This concept is modeled using differential equations, showing how the path of the pursuer can be determined based on the target's known path. This method is applicable beyond fictional scenarios, including missile guidance systems and aircraft.

Differential equations are crucial for systems with changing mass. The example of lifting a barbell with chains demonstrates how the effective mass changes as more chain leaves the ground. This complicates the equation of motion, requiring the use of the rate of change of momentum (F = d(mv)/dt) instead of the simpler F=ma. This principle is directly applicable to rocket science, where the rocket's mass decreases as fuel is expelled.

The SIR (Susceptible, Infected, Recovered) model for disease spread is a prime example of differential equations in action. This model tracks population categories and their transitions, showing how the rate of change for each category depends on the others. These equations help predict how an infection will spread through a population, with constants influenced by factors like social distancing. This model's principles are used by organizations like the International Council for Industrial and Applied Mathematics to understand real-world pandemics.

Differential equations are also used to model population growth, including complex interactions between different species. An example of bacteria multiplying and phages feeding on them illustrates how the rates of change for each population depend on multiples of the other. Phase portraits can be used to visualize these interactions and understand long-term behavior without explicitly solving the equations.

The video concludes by recommending Brilliant's differential equations courses, which focus on real-world applications like pursuit curves, wave equations, and beam behavior. These courses cater to various skill levels, offering a comprehensive understanding of differential equations and their practical uses.