Method for finding the complementary function
 In finding the complementary function, R.H.S. of the given equation is replaced by zero.
 Let y = C_{1} e^{mx} be the C.F. of d^{2}y/dx^{2} + P dy/dx + Qy = 0.
Put the values of y, dy/dx and d^{2}y/dx^{2} then C_{1}e^{mx}(m^{2} + Pm + Q) = 0 m^{2} + Pm + Q = 0 is called Auxiliary Equation.
 Solve the auxiliary equation.
 Roots real and different: If m_{1} and m_{2} are the roots, then the C.F. is
y = C_{1}e^{m1x} + C_{2}e^{m2x}
 Roots real and equal: If both the roots are m_{1} and m_{1}, then the C.F. is
y = (C_{1} + C_{2}x)e^{m1x}
 Roots Imaginary: If the roots are α ± iβ, then the C.F. is
y = e^{αx}[A cos βx + B sin βx]
