Method of variation of parameters

Variation of parameters is a more general way of solving second-order, linear, non-homogenous differential equation; i.e. of the form

𝑦 ″ + 𝑏 𝑦' + 𝑐 𝑦 = 𝑔 (𝑥)

Variation of parameters concerns the discovery of a particular solution, which can then be combined with the general solution of the complimentary homogenous equation. If the general solution is of the form

𝑦 𝑐 (𝑥) = 𝑘 1 𝑦 1 (𝑥) + 𝑘 2 𝑦 2 (𝑥)

where $𝑘 𝑖$ are constants and $𝑦 𝑖$ are functions of $𝑥$ , the particular solution involves varying the constants as functions

𝑦 𝑝 (𝑥) = 𝑢 (𝑥) 𝑦 1 (𝑥) + 𝑣 (𝑥) 𝑦 2 (𝑥)

which are given by the Wronskian

𝑢' (𝑥) = - 𝑦 2 (𝑥) 𝑔 (𝑥) 𝑊 [𝑦 1, 𝑦 2] 𝑣' (𝑥) = 𝑦 1 (𝑥) 𝑔 (𝑥) 𝑊 [𝑦 1, 𝑦 2]

This is in fact just an application of Cramer’s rule.

Generalisation to higher orders

A good explanation of the generalisation is given by this LibreText

Practice problems

2017. Elementary differential equations and boundary value problems, pp. 146–147 (§3.6 problems)

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Table of Contents

Method of variation of parameters

Generalisation to higher orders

Practice problems

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