Ordinary differential equations Linear equations and systems Proposition

Superposition principle

Builds on Homogeneous linear equation Linear differential equation of order n

Statement

Let $I$ be an open interval, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$ with continuous functions $a_0, \dots, a_{n-1}$ on $I$ . Let $y_1$ and $y_2$ be solutions of the homogeneous linear equation $L y = 0$ on $I$ , and let $c_1, c_2$ be real numbers. Then $c_1 y_1 + c_2 y_2$ is a solution of $L y = 0$ on $I$ .

Remarks

The proof is the linearity of $L$ : $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2 = 0$ . The same argument gives a variant for right sides: if $L y_1 = b_1$ and $L y_2 = b_2$ , then $L(y_1 + y_2) = b_1 + b_2$ . The principle fails for nonlinear equations. For $y' = y^2$ , the function $y = -1/x$ is a solution on $x > 0$ , but $2y$ is not.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Order of a differential equation definition
Linear differential equation of order n definition
Homogeneous linear equation definition
Superposition principle statement

Tag	`0F2VBPXNP9`
URI	`m2://0F2VBPXNP9`
Kind	statement, proposition
Writing	W0 stub
MSC	34A30
Status	This copy of the site has no build data. Run `m2 status 0F2VBPXNP9`.
Uses	Homogeneous linear equation, Linear differential equation of order n
Same as	Wikidata Q284885 Wikipedia Superposition principle

Superposition principle

Statement

Remarks

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