 Hi and welcome to the session. Let us discuss the following question. Question says, for the exercise given below, verify that the given function is a solution of the corresponding differential equation. This is the given function and this is the corresponding differential equation. Let us now start with the solution. Now given function is x y is equal to a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x plus x square. Now let us name this expression as 1. Now differentiating both sides of expression 1 with respect to x, we get x multiplied by dy upon dx plus y multiplied by 1 is equal to a multiplied by e raised to the power x minus b multiplied by e raised to the power minus x plus 2x. We can find derivative of this term by using product rule. Here also we can find derivative of this term by using product rule. Similarly we can find derivative of this term and we know here we will apply this formula to find derivative of x square replacing n by 2 in this formula we get derivative of x square is equal to 2x. Now let us name this equation as 2. Now we will again differentiate this equation 2 with respect to x. Now equation 2 becomes x multiplied by d square y upon dx square plus dy upon dx multiplied by 1 plus dy upon dx. We know we can find derivative of this term by using product rule and these two terms represent derivative of this term. Derivative of y is dy upon dx. Now we will write this is equal to sign as it is. Derivative of a multiplied by e raised to the power x is a multiplied by e raised to the power x only. Now we will write this minus sign as it is and derivative of b multiplied by e raised to the power minus x is minus b multiplied by e raised to the power minus x. Derivative of 2x is 2 so here we can write plus 2. Now simplifying further we get x multiplied by d square y upon dx square plus 2 dy upon dx is equal to a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x plus 2. From equation 1 clearly we can see we can find value of a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x. So we can write from 1 we get a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x is equal to x y minus x square. Subtracting x square from both the sides of this equation we get a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x is equal to x y minus x square. Now let us name this equation as equation 3 and this equation as equation 4. Now we will substitute x y minus x square for some of these two terms in equation 3. So we can write substituting equation 4 in equation 3 we get x multiplied by d square y upon dx square plus 2 multiplied by dy upon dx is equal to x y minus x square plus 2. Now this further implies x multiplied by d square y upon dx square plus 2 dy upon dx minus x y plus x square minus 2 is equal to 0. Now clearly we can see this equation is same as the given differential equation. So we can write the above equation is same as the given differential equation. Hence verified that the function x y is equal to a multiplied by e raised to the power x plus b multiplied by e raised to the power minus x plus x square is a solution of the given differential equation. This completes the session. Hope you understood the solution. Take care. Have a nice day.