 Differential equations are one of the most important areas of applied mathematics, so let's start talking about differential equations. The importance of differential equations emerges as follows. We often want to know the value of a quantity, but we find the change of the quantity easier to measure. So velocity is hard to measure, but acceleration is very easy. Population is hard to measure, but birth and death rates are easy. Or we might be able to describe the rate of change more easily than the amount of the quantity. So, for example, in Newton's law of cooling, the rate of change of cooling is proportional to the difference in temperature between an object and its environment. Or the law of radioactive decay, the number of radioactive decays is proportional to the number of atoms. Now, in calculus, you might remember that we use the derivative to describe rates of change. And so we have a differential equation is an equation that involves a derivative. Now, we classify differential equations according to a number of different terms. One of the more important ones is ordinary. An ordinary differential equation is an equation that involves some single variable function f and its derivatives. The order of the differential equation is the highest derivative included in the equation. So we're about to throw down some derivatives. So it's useful to remember that we have many, many, many different ways of expressing the derivative of a function. Suppose y is a function of t. We can represent the derivative of y with respect to t as y prime of t, dy over dt. Here's one you'll see a lot in physics and in engineering, this so-called dot notation. We have Newton to blame for that. y with a dot over it also represents a derivative with respect to time. And we sometimes use index notation y sub t as a representation of the derivative of y with respect to t. And there's more ways of doing that, but these are probably the four most common. And unfortunately, we switch between these with no real rhyme or reason. So for example, f prime of x equals f of x. Well here, f is a single variable function. So this is a ordinary differential equation. And the highest derivative is the first derivative. So this is a first order differential equation. d5y over dx to the fifth plus y equals x. If we assume that y is a single variable function of x, which is implied by the fact that we're differentiating with respect to x, the highest order derivative here is the fifth. So this is a fifth order differential equation. Or double dot y plus ty plus dot y dot y equals zero. When we use the dot notation, we almost always imply that we're taking the derivative with respect to t and that our functions are functions of t. So this is still ordinary. And this double dot is a second derivative. So this is a second order differential equation. So let's talk standards. The standard form of an nth order differential equation involves finding that nth derivative as a function of our independent variables and all of the lower order derivatives. In other words, it's the equation you get when you solve for the nth derivative. As a general rule, we don't actually care about writing equations in standard form. However, it will be useful to refer to the standard form when discussing differential equations in general. So put in standard form, the differential equation, and, well, who cares? Okay, so from time to time somebody may ask you to put in a differential equation in standard form and from time to time it'll actually be useful to do so. So we want to solve for the highest order derivative. So this is a first derivative and this is a third derivative. So we'll solve for this third derivative. And again, our result is not particularly interesting or useful, but it is something that we can work with. Now, we've talked about ordinary differential equations, so maybe there are some not-so-ordinary differential equations. And the distinction between the two is that we've assumed our function is a single variable function, but most functions aren't. So, for example, the area of a rectangle is a function of its length and width. Or, here's an important one, the temperature at a point on a heated plate is a function of its x and y coordinates. And, of course, the votes of a politician are functions of the desires of their constituents. And you know from calculus that if you have a multivariable function, you can still talk about its derivatives in the form of a partial derivative. And so a partial differential equation is a differential equation involving the derivatives of a multivariable function. And we name them the same way. So in this differential equation, the highest order derivative is a first partial derivative, and so this is a first order partial differential equation. And this differential equation, the highest order derivative, is a second derivative. So this is a second order partial differential equation. One particularly important distinction is linear. So remember that we had a standard form of a differential equation that was mostly useful when we could talk about the differential equations. And so remember the standard form arises when we solve for the highest order derivative. And we say that a differential equation is linear if f is linear in the function and its derivatives. So for f' of x equals f of x, we still have a first order differential equation, but because we have our function and its derivatives all appearing to the first power only, this is also a linear differential equation. Similarly, this is still a fifth order differential equation, but again our function and its derivatives only appear to the first power, and so this is a linear differential equation. And for this one, well here we have a derivative multiplied by another derivative, and this breaks the linearity, and so this is a second order nonlinear differential equation. It's much, much, much, much, much harder to solve nonlinear differential equations, so we'll do those later. And one last note, differential equations is a field of mathematics where some of the descriptors become very extensive, first order linear non-homogeneous partial differential equation. And so when we talk about differential equations we tend to use a lot of shortcut phrases. Your street cred as a mathematician depends on how well you sling phrases like ODE, ordinary differential equations, PDE, partial differential equations, Diffie Q, differential equation. Dear God, no, that's any nonlinear differential equation.