Complementary Function

Method for finding the complementary function
  1. In finding the complementary function, R.H.S. of the given equation is replaced by zero.

  2. Let y = C1 emx be the C.F. of d2y/dx2 + P dy/dx + Qy = 0.
    Put the values of y, dy/dx and d2y/dx2 then
    C1emx(m2 + Pm + Q) = 0
    m2 + Pm + Q = 0 is called Auxiliary Equation.

  3. Solve the auxiliary equation.
    1. Roots real and different: If m1 and m2 are the roots, then the C.F. is
      y = C1em1x + C2em2x
    2. Roots real and equal: If both the roots are m1 and m1, then the C.F. is
      y = (C1 + C2x)em1x
    3. Roots Imaginary: If the roots are α ± iβ, then the C.F. is
      y = eαx[A cos βx + B sin βx]


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