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How to solve d^2y/d^x2=logXPlease answer me soonI am waiting

How to solve d^2y/d^x2=logXPlease answer me soonI am waiting

Grade:12

1 Answers

Nandana
110 Points
7 years ago
hi ,
      given ,
              
                d/dx(dy/dx) = log x
                by integrating  on both sides , we get 
                dy /dx = ∫ (log x) dx
                           = x (log x -1 ) + C  [ by using ∫u dv = uv – ∫ v du , where u =log x & dv = dx ]
                  again integrate on both sides ,  we get
                           y  = ∫x logx dx – ∫x dx + c∫ dx  ------------- (1)
                  ∫ x log x dx  is given as
                    u =  x , dv = log x dx 
                    du = dx , v = ∫ logx dx = x (log x -1 )
                   ∫ x log x dx = x^2 (log x -1 ) – ∫ x logx dx + ∫ x dx
                                       = ½ ( x^2( log x -1 ) + ∫ x dx )  ----------- (2)
                     put  (2) in (1)
                 we obtain ,
                      y =  ½ (x^2 (log x-1 )) – ½ ∫ x dx + ∫ x dx + K
                 finally , we have
                        
                     y= ½ (x^2 (log x -1)) – ¼ x^2 + ½ x^2 + K
 

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