Question

Q. If the roots of an auxiliary equation are 1 and 1,
then the complementary
function is

Answer 1

Answer:

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]