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The solution of the differential equation ( d y d x ) y= (x+y)+ (x-y) is

Options

  1. Ax=-2 y+C
  2. By=-2 x+C
  3. Cy=2 y+C
  4. Dx=-2 y+C

Correct answer

B. y=-2 x+C

Step-by-step solution

The given differential equation is dy dx y = (x+y) + (x-y) . Using the trigonometric identity (A+B) + (A-B) = 2 A B , the equation becomes: dy dx y = 2 x y . Rearranging the terms to separate the variables: y y dy = 2 x dx . This simplifies to: y ^2 y dy = 2 x dx . Integrating both sides: y ^2 y dy = 2 x dx . Let u = y , then du = - y dy . The integral becomes: -u⁻² du = 2 x dx . u⁻¹ = -2 x + C . Substituting back u = y : 1 y = -2 x + C . y = -2 x + C . Answer: y=-2 x+C

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