Standard Form 
A linear homogeneous ordinary differential equation with constant coefficients has the general form of where are all constants. 
2nd Order Linear Homogeneous ODE with Constant Coefficients  
A second order linear homogeneous ordinary differential equation with constant coefficients can be expressed as This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable , since constant coefficients are not capable of correcting any irregular formats or extra variables. An elementary function which satisfies this restriction is the exponential function . Substitute the exponential function into the above differential equation, the characteristic equation of this differential equation is obtained This characteristic equation has two roots and .

nth Order Linear Homogeneous ODE with Constant Coefficients  
Similar to the second order equations, the form, characteristic equation, and general solution of order linear homogeneous ordinary differential equations are summarized as follows:
