Differential Equations

Here is the fifty-seventh episode of Quantum Foam, Differential Equations. We are going over this subject because it goes hand and hand with the idea of the last episode on mechanics. The subjects included in this decimal of the show are put together loosely. In mathematics, a differential equation is an equation that relates 1 or more known function and their derivatives. The functions generally represent physical quantities, the derivatives represent their rates of change, and a differential equation defines the relationship between the 2. The study of differential equations consists mainly of the study of their solutions, the set of functions that satisfy each equation, and the properties of their solutions. Jacob Bernoulli proposed the Bernoulli Differential Equation in 1695. Lagrangian mechanics is an alternate formulation of classical mechanics founded on the D'Alembert Principle. In this mechanics, there is a kinetic and potential energy of the system. In 1822, Fourier published his work on heat flow, the Analytic Theory Of Heat, basing his reasoning on Newton's law of cooling. There was a heat equation for conductive diffusion of heat. This idea is now part of academic curriculums. In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these ideas to be expressed dynamically, given the position, the velocity of an object, and all the various forces acting upon the body. An equation of motion can be solved explicitly.
A differential equation is an equation with a function and 1 or more of its derivatives. There are many tricks to solving these equations, if they can be solved at all. A common equation is 1 with a function and a derivative of that function. Differential equations can describe how populations can change, how heat moves, how springs vibrate, how radioactive material decays, and much more. The differential equation says it well but is hard to use. It can be solved using a special method called separation of variables. In physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. Over the years, wise people have worked out special methods to solve some types of differential equations. There are tell-tale signs that give away what kind of differential equation is present in our physics. We wouldn't use a rocket to go to the grocery store, and we can't walk to Mars. Ordinary Differential Equations have a single independent variable. Partial Differential Equations have 2 or more independent variables. The order is the highest derivative. The degree is the exponent of the highest derivative. Be careful not to confuse order with degree. When it is a d squared, it is an order 2 second derivative. An equation is linear when the variable and its derivative has no exponent or other function put on it. The first order linear differential equation has a linear polynomial in the unknown function and its derivatives. This is a good start on describing the fundamentals of what can be included in the study of Differential Equations.