 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 start with the solution. Now the given function is y is equal to x sin 3x. Let us name this equation as equation 1. Now differentiating both the sides of equation 1 with respect to x we get dy upon dx is equal to x multiplied by cos 3x multiplied by 3 plus sin 3x multiplied by 1. We know derivative of y is dy upon dx and here we have applied product rule to find the derivative of this term. Now this further implies dy upon dx is equal to 3x cos 3x plus sin 3x. Let us name this equation as equation 2. Now differentiating both the sides of equation 2 with respect to x again we get d square by upon dx square is equal to 3 multiplied by minus 3x sin 3x plus 3 cos 3x multiplied by 1 plus 3 cos 3x. We know derivative of dy upon dx is d square y upon dx square. And we can find derivative of this term by using product rule and derivative of sin 3x is 3 cos 3x. Now this further implies d square y upon dx square is equal to multiplying 3 by this term we get minus 9x sin 3x plus 3 cos 3x plus 3 cos 3x is equal to 6 cos 3x. Now from 1 we know y is equal to x sin 3x. So we can substitute y for x sin 3x here and we get d square y upon dx square is equal to minus 9y plus 6 cos 3x. Now adding 9y minus 6 cos 3x on both the sides of this equation we get d square y upon dx square plus 9y minus 6 cos 3x is equal to 0. Now this equation is same as the given differential equation. So we get given function is a solution of the given differential equation. Now we can write the above equation is same as the given differential equation. Therefore given function is a solution of the given differential equation. Hence verify this completes the session. Hope you understood the solution. Take care and keep smiling.