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AP EAMCET201921 Apr 2019Evening ShiftMathematicsDifferential EquationsActual

The differential equation formed by eliminating a and b from the equation y = e x ( a cos x + b sin x ) is

Options

  1. A2 d 2 y d x 2 + d y d x - 2 y = 0
  2. Bd 2 y d x 2 + 2 d y d x - 2 y = 0
  3. C2 d 2 y d x 2 - d y d x + 2 y = 0
  4. Dd 2 y d x 2 - 2 d y d x + 2 y = 0

Correct answer

D. d 2 y d x 2 - 2 d y d x + 2 y = 0

Step-by-step solution

The differential equation is given as, y = e x ( a cos x + b sin x ) Differentiate the above equation, d y d x = e x ( a cos x + b sin x ) + e x ( - a sin x + b cos x ) d y d x = y + e x ( - a sin x + b cos x ) ......(I) Again differentiate the above equation, d 2 y d x 2 = d y d x + e x ( - a sin x + b cos x ) + e x ( - a cos x - b sin x ) d 2 y d x 2 = d y d x - e x ( a cos x + b sin x ) + d y d x - y d 2 y d x 2 = 2 d y d x - y - e x ( a cos x + b sin x ) d 2 y d x 2 = 2 d y d x - y - y d 2 y d x 2 - 2 d y d x +

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