A Linear Homogeneous Second-Order Differential Equation with Constant Coefficients
This Demonstration shows how to solve a linear homogeneous differential equation with constant coefficients , where and are constant. First solve the characteristic equation . If and are two real roots of the characteristic equation, then the general solution of the differential equation is , where and are arbitrary constants. If , the general solution is . If , the general solution is .
The homogeneous linear differential equation
where is a function of , has a general solution of the form
where , , ..., are linearly independent particular solutions of the equation and , , …, are arbitrary constants.
If the coefficients , , …, are constant, then the particular solutions are found with the aid of the characteristic equation
To each real root of the characteristic equation of multiplicity , there corresponds particular solutions , , …, .
To each pair of imaginary roots of multiplicity , there corresponds pairs of particular solutions
 V. P. Minorsky, Problems in Higher Mathematics, Moscow: Mir Publishers, 1975 p. 261.