The Wronskian of a pair of differentiable functions f(x),...

The Wronskian of a pair of differentiable functions $f(x), g(x)$ is the scalar function $$ W[f(x), g(x)]=\operatorname{det}\left(\begin{array}{rr} f(x) & g(x) \\ f^{\prime}(x) & g^{\prime}(x) \end{array}\right)=f(x) g^{\prime}(x)-f^{\prime}(x) g(x) . $$ (a) Prove that if $f, g$ are linearly dependent, then $W[f(x), g(x)] \equiv 0$. Hence, if $W[f(x), g(x)] \not \equiv 0$, then $f, g$ are linearly independent. (b) Let $f(x)=x^3, g(x)=|x|^3$. Prove that $f, g \in \mathrm{C}^2$ are twice continuously differentiable and linearly independent, but $W[f(x), g(x)] \equiv 0$. Thus, the Wronskian is not a fool-proof test for linear independence. Remark. It can be proved, [7], that if $f, g$ both saticfy a secend order linear ordinary differential equation, then Page 123 ir 702 everdent @ And +ly if $W[f(x), g(x)] \equiv 0$.

The Wronskian of a pair of differentiable functions $f(x), g(x)$ is the scalar function
$$
W[f(x), g(x)]=\operatorname{det}\left(\begin{array}{rr}
f(x) & g(x) \\
f^{\prime}(x) & g^{\prime}(x)
\end{array}\right)=f(x) g^{\prime}(x)-f^{\prime}(x) g(x) .
$$
(a) Prove that if $f, g$ are linearly dependent, then $W[f(x), g(x)] \equiv 0$. Hence, if $W[f(x), g(x)] \not \equiv 0$, then $f, g$ are linearly independent. (b) Let $f(x)=x^3, g(x)=|x|^3$. Prove that $f, g \in \mathrm{C}^2$ are twice continuously differentiable and linearly independent, but $W[f(x), g(x)] \equiv 0$. Thus, the Wronskian is not a fool-proof test for linear independence.
Remark. It can be proved, [7], that if $f, g$ both saticfy a secend order linear ordinary differential equation, then Page 123 ir 702 everdent @ And +ly if $W[f(x), g(x)] \equiv 0$.

Key Concepts

Wronskian

The Wronskian is a determinant constructed from a set of functions and their derivatives. For two functions, it is defined as the determinant of a 2×2 matrix with the functions in the first row and their derivatives in the second row. This scalar function is used to examine the linear independence of functions, particularly in the context of solutions to differential equations.

Linear Independence and Dependence

Two functions are said to be linearly independent if one cannot be expressed as a constant multiple or a linear combination of the other over an interval. Conversely, they are linearly dependent if one function can be written as a constant multiple of the other. The Wronskian is a tool used to test this: if the Wronskian is nonzero at some point, the functions are linearly independent, while a zero Wronskian is a necessary (but not always sufficient) condition for linear dependence.

Continuous Differentiability

Continuous differentiability refers to functions having continuous derivatives up to a certain order. When working with the Wronskian, it is essential that the functions involved are sufficiently smooth (i.e., they belong to an appropriate C^n class) so that the derivatives exist and are continuous. This smoothness is crucial for the valid application of the Wronskian in analyzing linear independence and for ensuring the proper behavior of solutions to differential equations.

Role in Differential Equations

In the study of linear ordinary differential equations, the Wronskian plays an important role. If a set of functions are known to be solutions of a linear differential equation, then abnormally vanishing Wronskians can indicate issues with the solution set. Specifically, a nonzero Wronskian at a point guarantees that the functions form a fundamental set of solutions, although under certain conditions a zero Wronskian does not necessarily imply linear dependence.

Limitations and Counterexamples

There exist cases where functions are linearly independent even though their Wronskian is identically zero. These counterexamples highlight the limitations of using the Wronskian as the sole test for linear independence without additional conditions. Such scenarios typically occur when the functions do not satisfy certain analyticity or differentiability conditions required for the classical equivalence between a zero Wronskian and linear dependence to hold.

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