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Sunday, April 5, 2009

Working Rules For Finding Complementary Function of Linear Differential Equation

Case 1: -

If the roots are unequal (m = m1, m2, m3) then the complementary function is

C.F = c1em1x + c2em2x + c3em3x

Case 2: -

If the roots are equal (m = m1, m1, m1) then the complementary function is

C.F = (c1 + c2x + c3x2) em1x

Case 3: -

If the roots are complex (m = a ± ib) then the complementary function is

C.F = eax (c1cos bx + c2 sin bx), c1eax cos(bx + c2) or, c1eax sin (bx + c2)

And if the two equal part of complex roots (m = a ± ib, a ± ib) then the complementary function is

C.F = eax {(c1 + c2x) cos bx + (c3 + c4x) sin bx}

Case 4: -

If the roots are “a ± √b” then the complementary function is

C.F = eax (c1cos x√b + c2 sin x√b), c1 eax cosh (x√b + c2) or, c1 eax sinh (x√b + c2)