 Hi and welcome to the session I am Shashi and I am going to help you with the following question. Question is if y is equal to 500 multiplied by e raised to the power 7x plus 600 multiplied by e raised to the power minus 7x show that d square y upon dx square is equal to 49y. Let us start the solution now. It is given that y is equal to 500 multiplied by e raised to the power 7x plus 600 multiplied by e raised to the power minus 7x. Now differentiating both sides with respect to x we get dy upon dx is equal to 500 multiplied by 7 e raised to the power 7x plus 600 multiplied by minus 7 e raised to the power minus 7x now this implies dy upon dx is equal to 3500 multiplied by e raised to the power 7x plus minus 4200 e raised to the power minus 7x. Now we will again differentiate both sides of this expression. And we can write again differentiating with respect to x we get d square y upon dx square is equal to 3500 multiplied by 7 e raised to the power 7x plus minus 4200 multiplied by minus 7 e raised to the power minus 7x. Now this is equal to 24500 multiplied by e raised to the power 7x plus 29400 multiplied by e raised to the power minus 7x. So second derivative d square y upon dx square is equal to 24500 e raised to the power 7x plus 29400 e raised to the power minus 7x. Now taking 49 as a common factor we get d square y upon dx square is equal to 49 multiplied by 500 e raised to the power 7x plus 600 e raised to the power minus 7x. We know that 500 e raised to the power 7x plus 600 multiplied by e raised to the power minus 7x is equal to y. So bracket is equal to y. So we will substitute for this bracket y. So we get d square y upon dx square is equal to 49 y as 500 multiplied by e raised to the power 7x plus 600 multiplied by e raised to the power minus 7x is equal to y. So we were required to prove this only. So we can write d square y upon dx square is equal to 49 y. Hence this completes the session. Hope you understood the session. Take care and goodbye.