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})