 Hello and welcome to the session. I am Deepika here. Let's discuss a question which says form the general solution of the following differential equation x d y by dx plus 1 minus x plus x y part x is equal to 0 where x is not equal to 0. Let's start the solution. Now the given differential equation is d y by dx plus y minus x plus x y part x is equal to 0. Now to find the general solution of the differential equation x d y by dx plus y minus x plus x y part x is equal to 0. First we have to express this differential equation as a linear differential equation of the form d y by dx plus p y is equal to q where e and q are constants or functions of x of t. Now this differential equation can be written as x d y by dx plus y into 1 plus x or x is equal to x or 1 or dividing by x this equation can be written as d y by dx plus y into 1 plus x or x over x is equal to 1. Let us give this equation as number 1. Now this is a linear differential equation of the form d y by dx plus p y is equal to q here p is equal to 1 plus x or x over x q is equal to 1. Now we know that integrating factor is given by e raised to power integral of p d s or this is equal to e raised to power integral of 1 over x plus x into d s and this is equal to e raised to power log mod x plus log mod sin x. Again by using the law of logarithm we can write this as e raised to power log mod x sin x. This is again equal to mod x sin x because e raised to power log fx is equal to fx only. Now we are given in that question that x is not equal to 0. Now if we take x less than 0 then x sin x is positive and if we take x greater than 0 then also x sin x is positive. So we can take the integrating factor as x sin x that is mod of x sin x is equal to x sin x. Now on multiplying both sides of equation 1 by x sin x we get d y by dx into x sin x plus y into 1 plus x or x over x into x sin x is equal to x sin x. The left hand side of this equation is the differential of the function y into x sin x and this is equal to x sin x. Now on integrating both sides with respect to x we have integral of d by dx of y into x sin x dx is equal to integral of x sin x dx. This can be written as y into x sin x is equal to now we will integrate the right hand side by path. Let us take x as the first function and sin x as the second function. So y into x sin x is equal to minus x cos x minus integral of minus cos x dx y into x sin x is equal to minus x cos x plus sin x. It can be written as y is equal to minus x cos the general solution of the given differential equation is y is equal to 1 over x minus cos x plus c over x sin. 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.