Ordinary differential equations Elementary methods of solution Concept

Variation of parameters

Builds on Linear differential equation of order n Fundamental system of solutions Wronskian General solution of an inhomogeneous linear equation

Summary

Variation of parameters is a method to find a particular solution of an inhomogeneous linear equation from a fundamental system of the homogeneous equation. Let $p, q, f$ be continuous on an open interval $I$ . Let $(y_1, y_2)$ be a fundamental system of $y'' + p y' + q y = 0$ on $I$ , with Wronskian $W$ . One looks for $y = u_1 y_1 + u_2 y_2$ with the extra condition $u_1' y_1 + u_2' y_2 = 0$ . Then the equation $y'' + p y' + q y = f$ becomes $u_1' y_1' + u_2' y_2' = f$ . This linear system gives $u_1' = -y_2 f / W$ and $u_2' = y_1 f / W$ , so a particular solution is $y_p = -y_1 \int \frac{y_2 f}{W}\, dx + y_2 \int \frac{y_1 f}{W}\, dx.$

Remarks

The method works for each continuous right side, unlike the method of undetermined coefficients. Example: for $y'' + y = \tan x$ on $(-\pi/2, \pi/2)$ , with $y_1 = \cos x$ , $y_2 = \sin x$ and $W = 1$ , one gets $y_p = -\cos x \ln\left(\sec x + \tan x\right)$ . For first-order systems, the same idea gives the variation of constants formula.

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Differential equation concept
Function concept
Interval concept
Wronskian definition
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
Homogeneous linear equation definition
Superposition principle statement
Solution space of a homogeneous linear equation has dimension n statement
Fundamental system of solutions definition
General solution of an inhomogeneous linear equation statement
Variation of parameters concept

Where this leads

4 pages build on this one.

Tag	`0Y6TH5PQVS`
URI	`m2://0Y6TH5PQVS`
Aliases	`variation-of-constants`
Kind	concept
Writing	W1 stated
MSC	34A30
Uses	Linear differential equation of order n, Fundamental system of solutions, Wronskian, General solution of an inhomogeneous linear equation
Same as	Wikidata Q1205942 Wikipedia Variation of parameters

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