(a) Prove that the set of solutions to the homogeneous...

(a) Prove that the set of solutions to the homogeneous ordinary differential equation $u^{\prime \prime}-4 u^{\prime}+3 u=0$ is a vector space. (b) Write the solution space as the span of a finite number of functions. (c) What is the minimal number of functions needed to span the solution space?

Key Concepts

Homogeneous Differential Equation

A homogeneous differential equation is one in which every term is a function of the unknown function and its derivatives, and the equation is set equal to zero. In this context, the homogeneous ordinary differential equation implies that the zero function is a solution and that any linear combination of solutions will also be a solution.

Vector Space of Solutions

The set of all solutions to a linear homogeneous differential equation forms a vector space because it satisfies the axioms of vector addition and scalar multiplication. This is due to the underlying linearity of the equation, which guarantees that the sum of two solutions and any scalar multiple of a solution is also a solution.

Superposition Principle

The superposition principle is a fundamental property of linear homogeneous differential equations. It states that if two functions are solutions to such an equation, then any linear combination of these two functions is also a solution. This principle is what underpins the structure of the solution space as a vector space.

Span and Basis

The span of a set of functions is the collection of all linear combinations of those functions. In the context of the solution space of a differential equation, a basis is a set of linearly independent solutions such that every solution can be written as a linear combination of these basis functions. This conceptual framework is used to express the entire solution set in a compact and efficient form.

Dimension of the Solution Space

The minimal number of functions needed to span the solution space is given by the dimension of that space. For a second-order differential equation, like the one given, the solution space is two-dimensional, meaning that two linearly independent solutions are required to form a basis for the entire solution space.

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