Solving non-homogenous second order ODEs

Method of undetermined coëfficients

The method of undetermined coëfficients is a method for finding a particular solution to an ODE that involves taking a guess (Ansatz) of the particular solution based on the form of the non-homogenous term .

The Ansatz is then substituted into the ODE to determine the coëfficients. The following table shows functions and their corresponding guesses, where , are polynomials of order, determined and undetermined respectively.

Non-homogenous term	Ansatz
or
or

Note that a linear combination of such non-homogenous terms leads to a linear combination of their corresponding Ansätze. No term of the Ansatz can be a solution of the homogenous equation, if this is the case the Ansatz is multiplied by .¹

A special case occurs with Cauchy-Euler differential equations due to their equidimensional structure.

Practice problems

• 2017. Elementary differential equations and boundary value problems, pp. 141–142 (§3.5 problems)

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Footnotes

1. 2017. Elementary differential equations and boundary value problems, p. 139 (§3.5) ↩