 Hello, I am welcome to the session, I am Deepika here. Let's discuss a question which says, find the general solution of the following differential equation. x log x d y by d x plus y is equal to 2 over x log x. Let's start the solution. The given differential equation is x log x d y by d x plus y is equal to 2 over x log x. Now we know that to solve this equation, first we have to express this equation in the form of d y by d x plus p y is equal to q, where p and q are constants or functions of x only. So, on dividing this equation by x log x, we can divide by d x plus y into 1 over x log x is equal to 2 over x log x into 1 over x log x or this can be written as d y by d x plus y over x log x is equal to 2 over x cap. Let us give this equation as number 1. Now this is a linear differential equation of the form d y by d x plus p y is equal to q, here p is equal to 1 over x log x and q is equal to 2 over x cap. Now we have to find the integrating factor and we know that integrating factor is given by e raise to power integral of p d x. So, this is equal to e raise to power integral of 1 over x log x d x and this is equal to e raise to power log of log x and this is equal to log x only. So, our integrating factor is log x. Now we will multiply equation 1 by the integrating factor. So, on multiplying both sides of equation 1 by integrating factor which is equal to log x. So, we have d y by d x into log x plus y over x log x into log x and this is equal to 2 over x cap into log x. This can be written as d y by d x into log x plus y over x is equal to 2 over x cap into log x. Now left hand side of this equation is the differential of y log x. So, this equation can be written as d by d x of y log x is equal to 2 over x cap into log x. Now on integrating both sides with respect to x we have integral of d by d x of y log x into d x is equal to integral of 2 over x cap log x d x or y log x is equal to i plus c where i is equal to integral of 2 over x cap into log x d x. Now we will find the value of i that is we will solve this integral. Now on integrating by parts we have i is equal to let us take log x as the first function and 1 over x cap as the second function. So, we have i is equal to 2 into log x into integral of 1 over x cap d x minus integral of d by d x of log x into integral of 1 over x cap d x or i is equal to 2 into log x into minus 1 over x minus integral of 1 over x into minus 1 over x d x or this can be written as i is equal to 2 into minus log x over x plus integral of 1 over x cap d x or i is equal to 2 into minus log x over x minus 1 over x or this can be written as i is equal to minus 2 over x into log x plus 1. So, this is the value of i. Now we will substitute the value of i in the equation y log x is equal to i plus c. Let us give this equation as number 2. So, on substituting the value of i in equation 2 we have y log x is equal to minus 2 over x into log x plus 1 plus c helps the general solution of the given differential equation is y log x is equal to minus 2 over x into 1 plus log x plus c. So, this is our answer for the above question. I hope the solution is clear to you and you have enjoyed the session. Bye and have a nice day.