Differential Equations

Case 1: - If the roots are unequal (m = m 1 , m 2 , m 3 ) then the complementary function is C.F = c 1 e m1x + c 2 e m2x + c 3 e m3x Case 2: - If the roots are equal (m = m 1 , m 1 , m 1 ) then the complementary function is C.F = (c 1 + c 2 x + c 3 x 2 ) e m1x Case 3: - If the roots are complex (m = a ± ib) then the complementary function is C.F = e ax (c 1 cos bx + c 2 sin bx), c 1 e ax cos(bx + c 2 ) or, c 1 e ax sin (bx + c 2 ) And if the two equal part of complex roots (m = a ± ib, a ± ib) then the complementary function is C.F = e ax {(c 1 + c 2 x) cos bx + (c 3 + c 4 x) sin bx} Case 4: - If the roots are “a ± √b” then the complementary function is C.F = e ax (c 1 cos x√b + c 2 sin x√b), c 1 e ax cosh (x√b + c 2 ) or, c 1 e ax sinh (x√b + c 2 )