An nth order equation for, right? It's called a term, linear. If capital of f is linear in the variables y and y prime, and as derivative of y, okay? In other words, capital f is linear, independent variable y and any of its derivative, okay? In other words, it means the differential equation of 4 is linear if it can be written as, right here, an(x) times n derivative of y + an- 1(x) times n- 1 derivative of y + a1(x) times y prime + a0(x) times y = g(x). If you can write this given equation in the form of the equation 6, then we call the differential equation is linear. Here we note the two things for linear ordinary differential equations. All the coefficient, a0(x), a1(x), and an(x), they are coefficients of dependent variable y or its derivatives, right? All those coefficients must depends only on the independent variable x, okay? Moreover, all terms involving the dependent variable y or its derivative are of the first degree, okay? That means that the differential equation is a first order. Let's confirm these notions through the examples. On the other hand, we call any ordinary differential equation, which is not linear, we call it to be nonlinear ordinary differential equation, okay? For example, y double prime- 2xy prime + 2y = 0. This is the equation called the Hermite's Equation, which appears in the study of the harmony we'll see later in quantum mechanics. This is the second order line, can you see it? And the linear because all the coefficients of y or its derivatives must be a function of x which is really true here, right? So that this is a linear, second or the ordinary differential equation, right? Let's look at the second example. y(2-3x)dx + x(3y-1)dy = 0, this is a modelling equation for computation between two spaces, okay? I claimed that this is nonlinear and first order, because you can rely to this equation, right? You can rely to this equation as, okay, you can rely to this equation as x(3y- 1) and y prime + (2- 3x)y = 0, right, okay? So this is a first order because the highest of the derivative involved is y prime, okay? So that this is the first order. But this is nonlinear because the coefficient of y prime, okay, is not the function of x only, but is a function of x and y, right? x times the 3 y minus 1, right? So because of this term, okay, it is nonlineal. Let's look at the third example, y times 1 + y prime square = c. c is some constant, okay? This is the equation for brachistochrone problem, okay? This is the first order, Okay, right? And a nonlinear because, okay? The y prime, the power of a y prime is not 1 but 2, right, okay, so that this is nonlinear, okay? The last example, square root of 1-y times y double prime + 2xy prime = 0. This is called Kiddler's equation, and which arise for the study of the gas flow through the porous media, okay? It's a second order, right? Trivially, and it's nonlinear because the coefficient of y double prime is a function of a y, okay? So this is a nonlinear, okay?