Solution Space of Nonhomogeneous Linear Equation forms Affine Space

Theorem

Let $\EE$ be a nonhomogeneous linear ordinary differential equation.

The solution to $\EE$ forms an affine space.

Examples

Arbitrary Example

Let $S$ be the solution set to the linear second order ordinary differential equation:

$y' ' - y = 1$

Then:

$S$ is an affine space

the vector space $X$ is the solution set to the linear second order ordinary differential equation:

$y' ' - y = 0$

such that $S \times X \to S$ is the mapping which defines the affine space $S$.

Proof

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Sources

• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): affine space