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KCET2026MathematicsDifferential Equations

If x [3] y = (x + y)^n and x dy dx - y = 0 , then n =

Options

  1. A1
  2. B6 5
  3. C5 6
  4. D4 9

Correct answer

C. 5 6

Step-by-step solution

Given equation: x [3] y = (x + y)^n Taking natural logarithm on both sides: 1 2 x + 1 3 y = n (x + y) Differentiating with respect to x : 1 2x + 1 3y dy dx = n x + y (1 + dy dx ) It is given that x dy dx - y = 0 dy dx = y x Substituting dy dx = y x in the differentiated equation: 1 2x + 1 3y ( y x ) = n x + y (1 + y x ) 1 2x + 1 3x = n x + y ( x + y x ) 5 6x = n x n = 5 6 Answer: 5 6

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