 Hi and welcome to the session. Let us discuss the following question. Question says the general solution of the differential equation e raised to the power x dy plus y e raised to the power x plus 2x dx is equal to 0 is We have to choose the correct answer from A, B, C and D. Let us now start with the solution. Now the given differential equation is e raised to the power x dy plus y multiplied by e raised to the power x plus 2x multiplied by dx is equal to 0. Now this implies e raised to the power x dy is equal to minus y multiplied by e raised to the power x plus 2x dx. Now dividing both the sides of this equation by e raised to the power x dx we get dy upon dx is equal to minus y e raised to the power x plus 2x upon e raised to the power x. Now this further implies dy upon dx is equal to minus y minus 2x upon e raised to the power x. Now adding y on both the sides of this equation we get dy upon dx plus y is equal to minus 2x upon e raised to the power x. Now clearly we can see this equation is in this form where p is equal to 1 and q is equal to minus 2x upon e raised to the power x. Now we can write this equation as dy upon dx plus y is equal to minus 2x multiplied by e raised to the power minus x. We know 1 upon e raised to the power x is equal to e raised to the power minus x. Now this is a linear differential equation so first of all we will find integrating factor of this equation. We know integrating factor of this equation is equal to e raised to the power integral of p dx where p is the coefficient of y. Now in this equation value of p is 1 so integrating factor is equal to e raised to the power integral 1 multiplied by dx. Now using this formula of integration we can find this integral so integrating factor is equal to e raised to the power x. Now let us name this equation as equation 1. Now we know general solution of this linear differential equation is given by y multiplied by integrating factor is equal to integral of q multiplied by integrating factor dx plus c. Now in this equation we will substitute corresponding values of integrating factor and q. Now we can write solution of the given linear differential equation y multiplied by e raised to the power x is equal to integral of minus 2x e raised to the power minus x multiplied by e raised to the power x dx plus c. Here we have substituted e raised to the power x for integrating factor and minus 2x e raised to the power minus x for q. Now we know e raised to the power minus x multiplied by e raised to the power x is equal to 1. So here we can write y multiplied by e raised to the power x is equal to integral of minus 2x dx plus c. Now this further implies y multiplied by e raised to the power x is equal to minus 2 multiplied by integral of x dx plus c. Now using this formula of integration we can find this integral and we get y multiplied by e raised to the power x is equal to minus 2 multiplied by x square upon 2 plus c. Now here 2 and 2 will get cancelled and we get y multiplied by e raised to the power x is equal to minus x square plus c. Now adding x square on both the sides of this equation we get y multiplied by e raised to the power x plus x square is equal to c. So this is the required general solution of the given differential equation. Our correct answer is c. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.