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 $𝑔 (𝑡)$ .

𝐿 [𝑦] = 𝑦 ″ + 𝑝 (𝑡) 𝑦' + 𝑞 (𝑡) 𝑦 = 𝑔 (𝑡)

The Ansatz is then substituted into the ODE to determine the coëfficients. The following table shows functions and their corresponding guesses, where $𝑝 𝑛 (𝑥)$ , $𝑃 𝑛 (𝑥)$ $𝑄 𝑛$ are polynomials of order, determined and undetermined respectively.

Non-homogenous term $𝑔 (𝑥)$	Ansatz
$𝑝 𝑛 (𝑥)$	$𝑃 𝑛 (𝑥)$
$𝑝 𝑛 (𝑥) 𝑒 𝛼 𝑥$	$𝑃 𝑛 (𝑥)$
$𝑝 𝑛 (𝑥) s i n (𝛽 𝑥)$ or $𝑝 𝑛 (𝑥) c o s (𝛽 𝑥)$	$𝑃 𝑛 (𝑥) s i n (𝛽 𝑥) + 𝑄 𝑛 (𝑥) c o s (𝛽 𝑥)$
$𝑝 𝑛 (𝑥) 𝑒 𝛼 𝑥 s i n (𝛽 𝑥)$ or $𝑝 𝑛 (𝑥) 𝑒 𝛼 𝑥 c o s (𝛽 𝑥)$	$𝑒 𝛼 𝑥 [𝑃 𝑛 (𝑥) s i n (𝛽 𝑥) + 𝑄 𝑛 (𝑥) c o s (𝛽 𝑥)]$

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 $𝑥$ .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)

#state/tidy | #lang/en | #SemBr | #review

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

Method of undetermined coëfficients

Practice problems

Footnotes