 Hello and welcome to the session. In this session we shall discuss about the differential equations. An equation involving derivatives of the dependent variable with respect to independent variable or it can be independent variables also is known as differential equation. Consider the equation x into dy by dx plus y equal to 0. In this the independent variable is the variable x and the dependent variable is the variable y. As you can see this equation involves derivative of the dependent variable that is y with respect to the independent variable that is x. So this equation is a differential equation. Now when a differential equation involves derivatives of the dependent variable with respect to only one independent variable like an equation of this kind 2d to y by dx2 plus dy upon dx the whole cube equal to 0 then in that case the differential equation would be called ordinary differential equation. Next we shall discuss order of a differential equation. Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Consider the equation d to y by dx2 plus y equal to 0. This is a differential equation in which the highest derivative of second order is involved that is this. So the order of this differential equation is 2. Next is the degree of a differential equation. Basically degree of a differential equation is defined if it is a polynomial equation in its derivatives and so the definition of the degree of a differential equation is given as that degree of a differential equation is the highest power of the highest order derivative in it. Consider the same differential equation that we had discussed above d to y by dx2 plus y equal to 0 in this the highest order derivative is this the power of this highest order derivative is 1 as you can see. So we say that the degree of this differential equation is 1. An important point to remember is order and degree of a differential equation always positive integers. We discuss about the solution of a differential equation. A function which satisfies the given differential equation is called its solution. Let's consider the differential equation d to y by dx2 plus y equal to 0. The solution of this differential equation is a function phi that will satisfy this equation that when we substitute y equal to phi x in this differential equation then the LHS would be equal to the RHS and so this is the solution of this differential equation. This is also called the solution curve or you can also say integral curve of this differential equation. Consider the function given by y equal to phi x equal to a sin x plus b where this a and b belongs to R. Then this is the solution of this differential equation. As you can see this function phi consists of two arbitrary constants a and b so this is called the general solution of the given differential equation that is the solution which contains as many arbitrary constants the order of the equation is called the general solution of the given differential equation. Now if we give some particular values to a and b suppose that a be equal to 2 and b be equal to pi upon 4 then we get y equal to let it be phi 1 x equal to 2 sin of x plus pi by 4. When this function phi 1 and its derivative are substituted in this differential equation then LHS would be equal to the RHS and so we say that phi 1 is the solution of the given differential equation. As you can see this phi 1 does not contain any arbitrary constants instead it contains only particular values of the arbitrary constants a and b. So this is called the particular solution of the given differential equation that is a solution free from arbitrary constants is called particular solution of the given differential equation. Consider the differential equation d2y by dx2 minus dy by dx equal to zero we have to check if y equal to e raised to the power x plus 1 is the solution of this differential equation or not. We have y equal to e raised to the power x plus 1 this gives us dy by dx is equal to e to the power x from here we have d2y by dx2 is equal to e to the power x. Now we substitute these two values in this differential equation we get e to the power x minus e to the power x equal to zero that is zero equal to zero we have LHS is equal to the RHS and hence y equal to e to the power x plus 1 is a solution of the differential equation d2y by dx2 minus dy by dx equal to zero. Now let's discuss the formation of a differential equation to form a differential equation from a given function we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants try and find out the differential equation that will represent the family of all parallel straight lines y equal to 3x plus c where c belongs to R is an arbitrary constant. Now the family of straight lines is given by y equal to 3x plus c in this case as you can see we have just one arbitrary constant that is c so we differentiate this function just once we get dy upon dx is equal to 3 this represents the family of all parallel straight lines given by y equal to 3x plus c this is how we form a differential equation this completes the session hope you have understood what is a differential equation the order and degree of a differential equation solution of a differential equation and how do we form a differential equation.