Ordinary Differential Equations ODE Home General Terms First Order ODE Higher Order ODE Homogen. Const. Coeff. Non-homogen. C. C. Variable Coeff. ODE Particular Solution by Undetermined Coeff. Variation of Parameters Reduction of Order Inverse Operators Systems of ODE Sturm-Liouville Transform Methods Numerical Methods Resources Bibliography
 Method of Reduction of Order Consider a linear non-homogeneous ordinary differential equation with constant coefficients where are all constants and . Let the ODE can be rewritten as Since all coefficients are constants, the above equation can be factored into Thus, The particular solution can be obtained by repeated integration of these inverse differential operators. Pros and Cons of the Method of Reduction of Order: The method of reduction of order is very straightforward but not always easy to perform unless all are real numbers. In addition, n integrations in sequence are not convenient to check. Modification of the Method of Reduction of Order: By performing the partial fraction expansion, the sequential integration can be broken into the sum of a serial individual integrations, i.e., If are k repeated roots, the particular solution becomes