Find two functions that form an orthogonal basis for the...

Find two functions that form an orthogonal basis for the space of the solutions to the differential equation $y^{\prime \prime}-3 y^{\prime}+2 y=0$ under the $\mathrm{L}^2$ inner product on $[0,1]$.

Key Concepts

Orthogonal Bases

An orthogonal basis for a vector space is a set of basis vectors (or functions) that are mutually orthogonal with respect to the given inner product. In the context of function spaces, orthogonality means that the inner product of any two distinct basis functions is zero. An orthogonal basis simplifies many problems in analysis and differential equations, particularly when projecting functions onto the subspace spanned by these basis elements.

Inner Product Spaces and the L² Space

An inner product space is a vector space equipped with an inner product, a way to multiply functions to yield a scalar that conveys geometric notions like length and angle. In the context of function spaces, the L² space on an interval involves functions whose squares are integrable, and the inner product is typically defined as the integral of the product of two functions over that interval. This setting provides the framework for discussing concepts such as orthogonality and norm.

Gram-Schmidt Orthogonalization

The Gram-Schmidt process is a method for converting a set of linearly independent functions into an orthogonal (or orthonormal) set with respect to a specified inner product. In cases where a natural basis for the solution space of a differential equation is not orthogonal with respect to the L² inner product, the Gram-Schmidt procedure can be used to generate an orthogonal basis, thereby facilitating the analysis and computations in the space.

Differential Equations with Constant Coefficients

These are equations in which the coefficients of the derivatives are constants. The key to solving such equations is forming the characteristic (or auxiliary) equation whose roots determine the form of the general solution. The method involves converting the differential equation into an algebraic one and then using the nature of its roots (real, repeated, or complex) to construct fundamental solutions.

Solution Space of Differential Equations

The solution set of a linear homogeneous differential equation forms a vector space. For a second order equation, this space is two-dimensional if the equation is non-degenerate. Establishing a basis for this space means finding a set of two linearly independent functions whose linear combinations account for every possible solution of the equation.

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