 Hello and welcome to the session. In this session we discussed the following question that says if y is equal to 3 into cos of log x plus 4 into sin of log x then show that x square into d2y by dx2 plus x into dy by dx plus y is equal to 0. Let's move on to the solution now. We are given that y is equal to 3 into cos of log x plus 4 into sin of log x. We have to show x square into d2y by dx2 plus x into dy by dx plus y is equal to 0. We take this as equation 1. Next we would differentiate both sides of equation 1 with respect to x. So differentiating both sides of equation 1 with respect to x we get dy by dx is equal to 3 into now differential of cos x is minus sin x. So it would be minus sin of log x into differential of log x which is 1 upon x then plus 4 differential of sin x is cos x. So here it would be 4 into cos of log x into differential of log x which is 1 upon x. So this is dy by dx. So further we get x into dy by dx is equal to minus 3 sin of log x plus 4 cos of log x. Let this be equation 2. Now we differentiate equation 2 with respect to x. So differentiating equation 2 on both sides with respect to x we get, now applying the product rule on the left hand side that is x into dy by dx we get x into differential of dy by dx would be d2y by dx2 plus dy by dx into differential of x which is 1 is equal to minus 3 into now differential of sin x is cos x. So it would be cos of log x into differential of log x which is 1 upon x then plus 4 into now differential of cos x is minus sin x. So 4 into minus sin of log x into differential of log x which is 1 upon x. So this further gives us x into d2y by dx2 plus dy by dx is equal to minus 3 upon x into cos of log x minus 4 upon x into sin of log x. This further implies x into d2y by dx2 plus dy by dx is equal to minus common then 3 into cos of log x plus 4 into sin of log x and this whole divided by x. Now further we get x square into d2y by dx2 plus x into dy by dx that is we have multiplied this x on this left hand side. This is equal to minus of 3 into cos of log x plus 4 into sin of log x. Now from equation 1 we have y is equal to 3 into cos of log x plus 4 into sin of log x. So this means we have x square into d2y by dx2 plus x into dy by dx is equal to minus y from equation 1. Or you can say x square into d2y by dx2 plus x into dy by dx plus y is equal to 0. This is what we were supposed to prove so hence proved. This completes the session hope you have understood the solution of this question.