 Hi and welcome to the session. Let's work out the following question. The question says solve the differential equation d2y by dx2 is equal to log x given that x is equal to 1, y equals to 1 and dy by dx is equal to 1 So let us start with the solution to this question Here d2y by dx2 is equal to log x Now integrating both the sides with respect to x we have dy by dx is equal to integral log x dx This implies dy by dx is equal to x log x minus x plus a constant c1 because we know that integration of log x dx is x log x minus x Now we call this equation 1 Now we put x equal to 1 y equal to 1 and dy by dx equal to 1 in equation 1 Therefore 1 is equal to 1 log 1 minus 1 plus c1 This implies c1 is equal to 2 Thus the equation 1 will become dy by dx is equal to x log x minus x plus 2 Now again integrating both the sides with respect to x we have y is equal to integral x into log x dx minus integral x dx plus integral 2 dx plus another constant c2 This implies y is equal to Now here we again apply the product rule Where in this is the first function this is the second function So we have first function into integral of x dx minus integral of dy dx of log x into integral x dx Minus this will be x square by 2 plus 2x plus the constant c2 This is equal to x square by 2 into log x Minus integral 1 upon x into x square by 2 dx minus x square by 2 plus 2x plus the constant c2 This is equal to x square by 2 into log x minus x by 2 dx is x square by 4 Minus x square by 2 plus 2x plus c2 This implies that y is equal to x square by 2 log x minus 3 by 4 x square plus 2x plus the constant c2 and we call this equation 2 Now we put x equal to 1 and y equal to 1 in equation 2 We get 1 is equal to 1 by 2 log 1 minus 3 by 4 into 1 square plus 2 into 1 plus the constant c2 This implies c2 is equal to 1 minus 0 plus 3 by 4 minus 2 and this implies that c2 is equal to minus 1 by 4 Hence the required solution is y is equal to x square by 2 log x minus 3 by 4 x square plus 2x minus 1 by 4 So this is our answer to this question I hope that you understood the solution and enjoyed the session. Have a good day