Explain why the functions cosx, sinx form an orthogonal...

Explain why the functions $\cos x, \sin x$ form an orthogonal basis for the space of solutions to the differential equation $y^{\prime \prime}+y=0$ under the $\mathrm{L}^2$ inner product on $[-\pi, \pi]$.

Key Concepts

Solution Spaces of Linear Differential Equations

A linear differential equation such as a second?order homogeneous equation has a solution space that forms a vector space. This space is finite-dimensional—typically two-dimensional for second-order equations—meaning that any solution can be expressed uniquely as a linear combination of two linearly independent solutions.

Fundamental Set of Solutions

A fundamental set of solutions is a collection of linearly independent functions from which every solution to the differential equation can be derived by taking appropriate linear combinations. This set forms a basis for the solution space, ensuring that the general solution captures all possible behaviors.

Inner Product Spaces

Function spaces, especially the space of square-integrable functions (L² spaces), can be equipped with an inner product. This inner product, defined as an integral over a specific interval, allows one to measure angles and lengths (norms) between functions, thereby introducing geometric concepts such as orthogonality.

L² Inner Product

The L² inner product on an interval is defined by integrating the product of two functions over that interval. This framework is crucial in many areas of analysis because it provides a natural setting for concepts like convergence, projection, and the computation of coefficients in expansions.

Orthogonality and Orthogonal Bases

Two functions are said to be orthogonal if their inner product is zero. When a set of functions is both orthogonal and spans a function space, it constitutes an orthogonal basis. This property simplifies many computations, as the coefficients for expressing any function in the space can be efficiently determined using projections onto the basis elements.

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