 Hi and welcome to the session. I am Deepika here. Let's discuss a question which says from the general solution of the differential equation 1 plus x square into dy plus 2xygx is equal to or x dx where s is not equal to 0. Let's start the solution. The given differential equation is 1 plus x square into dy plus 2xy dx is equal to cortex dx. Now we will express this equation as a linear differential equation of the form dy by dx plus py is equal to q where p and q are constants or functions of x only or this can be written as 1 plus x square into dy by dx plus 2xy is equal to cortex. Here we have divided each term by dx. Again this can be written as dy by dx plus 2xy over 1 plus x square is equal to cortex over 1 plus x square. Let us give this equation as number 1. Now this is a linear differential equation of the form dy by dx plus py is equal to q. Here p is equal to upon 1 plus x square and q is equal to cortex upon 1 plus x square. Therefore integrating factor is given by e raise to power integral of p dx or we have this is equal to e raise to power integral of 2x over 1 plus x square into dx and this is equal to e raise to power law of 1 plus x square. Now again this is equal to 1 plus x square because e raise to power law of fx is equal to fx. So our integrating factor is 1 plus x square. Now we will multiply equation 1 by 1 plus x square. So on multiplying both sides of equation 1 by the integrating factor which is equal to 1 plus x square we have dy by dx into 1 plus x square plus 2xy over 1 plus x square into 1 plus x square and this is equal to cortex over 1 plus x square into 1 plus x square or this can be written as dy by dx into 1 plus x square plus 2xy is equal to cortex. Now clearly left hand side of this equation is a differential of the function y into 1 plus x square. So this equation can be written as dy dx of y into 1 plus x square is equal to cortex. Now integrating both sides with respect to x we have integral of dy dx of y into 1 plus x square dx is equal to integral of cortex dx or this can be written as y into 1 plus x square is equal to log of more sin x plus c or y is equal to 1 plus x square raised to power minus 1 log of more sin x plus c into 1 plus x square raised to power minus 1. Hence the general solution of the given differential equation is y is equal to 1 plus x square raised to power minus 1 into log of more sin x plus c into 1 plus x square raised to power minus 1. So this is our answer of the above question. I hope the solution is clear to you and you have enjoyed the session. Bye and have a nice day.