Okay, let's begin with chapter two. First order differential equations. First order differential equations are rather special compared to other high order differential equations, because many first order differential equations allow analytic methods of finding their general solutions, okay. So, here we'd like introduce a subclass of first order differential equations which allow analytic methods to obtain their general solutions. First, let me introduce the separable differential equation. A first order differential equation is called separable if it is of the form the y prime is equal to g of x times h of y, in other words, this right hand side is a product of function which is a function of only one variable x and another function h which is a function of only y variable, okay. For example, consider the y prime is equal to x minus 2 x y over y squared minus three. At first it may look like it's not separable, but in fact, okay, here we have a divided side which is the x minus two times of x times of y over y squared minus three, from these two terms x is common, so that you can write it as x times one minus 2 y, and over y squared minus three, so, this is really the separable, because here you have a function of x only and the remaining part one minus 2 y over y squared minus three is a function of y only, okay. So, the differential equation down there, this is a separable. On the other hand, the second example the y prime is equal two minus x squared minus y is not separable, because you cannot express this right hand side in the form of g x times h of y, okay. It's very simple to distinguish equation which is separable and the equation which is not separable. Okay. Let's introduce the another symbol that q over y equal to one over h of y, then our original equation, okay, if you remember that it was y prime is equal to g of x times h of y, right, now I said one of h of y is equal to q of y. In other words, the h of y is equal to one over q of y, okay. So, and the last of y to this y prime in life in notation say d y over d x, right, d y over d x, then multiply both sides by q y, and again multiply both sides by g of x, then you are going to get the following equation, okay, q of y times d y that is equal to g of x times of d of x, right? This last equation down there, the equation number three here, recorded as okay, we say that in this equation the variable is really separated, because on one side of the equality you have only y, and on the other side of equality you have only x, right? How to solve it? This is very simple, just taking integration on both sides, right. So take integration of both sides, right, then, from this left side the left denoted by the capital Q of y to be any antiderivative of little q of y, okay, then through this integration you might get into a constants of y, okay, and that should be equal to the integral of g over x d x then let's denote by the capital G of x to be any one of the antiderivative of little g of x, and then plus you might have another integrating factor c 2, okay, so, okay, you can write it again Q of y equals G of x and plus c 2 and minus c over 1, okay. What are these two numbers c 2 and c 1? They are just to integral constants and they can be arbitrary real number, okay. If c 2 and the both c 2 and the c 1 are arbitrary real numbers, their subtraction is again an arbitrary real number, so we can denote it simply by c, right. So, we get to this form down there, Q over y is equal to G of x plus arbitrary constant of c, okay. As I said, the capital Q y is any antiderivative of little q of y and the capital G of x is any antiderivative of little g of x, right. The equation three, okay, this expression down there is a general solution to the given differential equation, separable differential equation, y prime is equal to G x times h of y, okay. And this is the implicit form of this solution, because we do not solve this equation for y in terms of x, right.
