How to solve d^2y/d^x2=logXPlease answer me soonI am waiting
Yogesh , 9 Years ago
Grade 12
Differential Calculus> How to solve d^2y/d^x2=logXPlease answer ...
1 AnswersNandana
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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