 Alright friends, in this video we are going to talk about solution of differential equations. Now let us understand what do we mean by a solution of an equation first. So what is the solution of x plus 3 equal to 5, right, average of simple linear equation in a variable x. So what is the solution of this? You say solution of this is equal to 5 minus 3 which is 2, right. yx is equal to 2 is a solution because 2 satisfies this equation, isn't it? So in a similar way, the function or the curve or the family of curves which satisfies a differential equation will be called the solution of the differential equation, right. So if I say dy by dx is equal to x, right. What is the solution of this differential equation? The solution of this differential equation will be x square by 2, right or you can say x square by 2 plus c, right. Because if you find the derivative of y with respect to x, I am definitely going to get dy by dx as x, isn't it? So this is a solution for this differential equation or in other words, this differential equation has been formed from this family of curves, right. Now to tell you frankly, while you were doing integration chapter, you were actually already learning how to find solutions for differential equation, right. So when a question like this was posed to you, what is the integral of let's say 2x plus 5 with respect to dx, right. Then you integrated the answer to be x square plus 5x plus c, right. The same thing if I have to write in the form of a differential equation, I would say what is that curve whose derivative gives you 2x plus 5. So the answer will be the curve whose derivative gives me 2x plus 5 is nothing but y equal to x square plus 5x plus c, right where c is an arbitrary constant. So indirectly you are actually learning how to find solutions of differential equation. So whole integration chapter was nothing but solving or finding the solutions for differential equation of order 1, right. How do we know it's order 1? Because ultimately we are going to write c, one arbitrary constant involved and in other words we have only dy by dx term. So the highest differential coefficient occurring in that particular differential equation is 1. So in the integration chapter, particularly in definite integration, we were learning how to solve differential equations of order 1 or we were learning the solutions for that, okay. So in this video, my main aim is to tell you how many types of solutions are there for the differential equation. Alright, it's not talking about the type of solutions possible for a differential equation. Normally we speak of the solution to a differential equation are of two types. One is called the general solution and the other one is called the particular solution, right. Now what is the general solution? To explain you in words, general solution of a differential equation is that family of curves which contains the arbitrary constants, right. But however, if you involve some values or if you assign the values to those arbitrary constants, the general solution will convert itself to a particular solution, right. Let me explain this with the help of an example. If I say dy by dx is equal to 2x, right. So when we solve this differential equation, we solve this differential equation as y equal to x square plus c. So y equal to x square plus c is a solution for this differential equation. Now because I have written a c here which is an arbitrary constant and I don't know what is the value of this arbitrary constant, such a solution will be called the general solution, right. So in short, in the general solution c or arbitrary constants, we can say arbitrary constants would be present. So it would contain your arbitrary constants, right. That depends what is the order of the differential equation. So if it is a second order differential equation, it will contain two arbitrary constants. If it is a third order differential equation, it will contain three arbitrary constants. So since this is an order one differential equation, it will contain one arbitrary constant. So writing a solution where an arbitrary constant is written and I don't know what is the value of that arbitrary constant, then that solution will be called the general solution. On the other hand, the very same differential equation, if I say y is equal to 1 when x is equal to 0, right. So find the solution for this differential equation when y is equal to 1 and x is equal to 0. Then in order to find the solution, we will say the general solution is y equal to x square plus c, no doubt about that and then we will use this value. This value is called the initial values, right. Similarly in the terms of differential equation chapter, we call such information given to us as initial values. So these are called initial values. Now using these initial values, we can actually assign some value to this arbitrary constant. How? Use this information that when y is 1, x is 0 or when x is 0, y is 1. So when we put y as 1, x is equal to 0. This is the condition that we get, right, which implies your c is nothing but 1, right. So if we write the same equation as, if I write the same equation as y is equal to x square plus c, which is nothing but 1, then we are writing what? We are writing the particular solution. So we are writing the particular solution for that differential equation. So to sum up in few words, differential equation will have a general solution if I am writing the solution involving arbitrary constants, where the values of those arbitrary constants are not known. If a differential equation solution, we are writing by assigning some value to the arbitrary constant, that means we know what is the value of the arbitrary constant and how do we know that? We know it from something called initial values and that will be given to you in the question. So if you are writing the solution without involving any arbitrary constants or in other words we can say the same thing as, if your solution involves such arbitrary constants whose values are known to you, then that solution will be called the particular solution. So this is called the particular solution, let me write it, yes. And this is called the general solution, so this is your general solution, okay.