Question

If \(W(f, g)\) is the Wronskian of \(f\) and \(g,\) and if \(u=2 f-g, v=f+2 g,\) find the Wronskian \(W(u, v)\) of \(u\) and \(v\) in terms of \(W(f, g) .\)

Short answer

Question: Find the Wronskian of \(u = 2f - g\) and \(v = f + 2g\), in terms of \(W(f, g)\). Answer: \(W(u, v) = 3f\frac{dg}{dt} - 2W(f, g)\)

Step-by-step solution

Compute the derivatives of u and v

We are given the following functions: $$ u = 2f - g \\ v = f + 2g $$ Now, let's differentiate these functions with respect to time: $$ \frac{du}{dt} = 2 \frac{df}{dt} - \frac{dg}{dt} \\ \frac{dv}{dt} = \frac{df}{dt} + 2 \frac{dg}{dt} $$

Calculate the determinant of the matrix

To find the Wronskian \(W(u, v)\), we need to calculate the determinant of the matrix formed by \(u, u', v,\) and \(v'\). The matrix is: $$ \begin{pmatrix} u & v \\ \frac{du}{dt} & \frac{dv}{dt} \end{pmatrix} = \begin{pmatrix} 2f-g & f+2g \\ 2\frac{df}{dt}-\frac{dg}{dt} & \frac{df}{dt}+2\frac{dg}{dt} \end{pmatrix} $$

Apply the determinant formula

Now, let's calculate the determinant of the matrix: $$ W(u, v) = \begin{vmatrix} u & v \\ \frac{du}{dt} & \frac{dv}{dt} \end{vmatrix} = (2f-g)(\frac{df}{dt}+2\frac{dg}{dt}) - (f+2g)(2\frac{df}{dt}-\frac{dg}{dt}) $$

Simplify the expression

Expanding and simplifying, we get: \begin{align*} W(u, v) &= (2f\frac{df}{dt} + 4f\frac{dg}{dt} - g\frac{df}{dt} - 2g\frac{dg}{dt}) - (2f\frac{df}{dt} - f\frac{dg}{dt} + 4g\frac{df}{dt} - 2g\frac{dg}{dt}) \\ &= 2f\frac{dg}{dt} - g\frac{df}{dt} + f\frac{dg}{dt} - g\frac{df}{dt} \\ &= 3f\frac{dg}{dt} - 2g\frac{df}{dt} \end{align*}

Express \(W(u, v)\) in terms of \(W(f, g)\)

Recall that the Wronskian \(W(f, g)\) is given by: $$ W(f, g) = f\frac{dg}{dt} - g\frac{df}{dt} $$ Multiplying both sides of the equation by \((-2)\), we get: $$ -2W(f, g) = -2f\frac{dg}{dt} + 2g\frac{df}{dt} $$ Now, in order to find an expression for \(W(u, v)\) in terms of \(W(f, g)\), we need to add \(3f\frac{dg}{dt}\) to both sides of the equation: $$ W(u, v) = 3f\frac{dg}{dt} - 2W(f, g) $$ Now, we have found \(W(u, v)\) in terms of \(W(f, g)\).

Key concepts

Determinant of a Matrix

In mathematics, a matrix is a rectangular array of numbers arranged in rows and columns. The determinant of a square matrix is a unique number that can be derived from its elements. It plays a crucial role in various applications of linear algebra.
The determinant gives important information about the matrix, such as whether it is invertible (a non-zero determinant means it has an inverse) and the scaling factor of the linear transformation represented by the matrix.
In the case of a 2x2 matrix, which we often use when calculating the Wronskian, the determinant can be calculated using the formula:

• If the matrix is \(\begin{pmatrix}a & b \ c & d \ \end{pmatrix}\)", then the determinant is \((ad - bc)\).

This formula helps us understand how the elements of the matrix interact to produce the determinant.
Using determinants, we can analyze the linear independence of functions, which is a core element when discussing the Wronskian.

Differentiation

Differentiation refers to the mathematical process of finding a derivative, which measures how a function changes as its variable changes. It is a fundamental concept in calculus and vital for understanding rates of change, like velocity.
The derivative of a function gives us the slope of the tangent line at any given point. In context with Wronskian, we differentiate functions to examine their rates of change relative to each other.
When calculating the Wronskian, we need the derivatives of the involved functions to construct the matrix. For example, if given functions are \(f(t)\) and \(g(t)\), after differentiating, we get \(\frac{df}{dt}\) and \(\frac{dg}{dt}\), which are then used to form the matrix for which we calculate the determinant.
Differentiation in such cases reveals if functions are linearly independent, by showing whether their Wronskians are non-zero.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and the linear mappings between them. This field includes the study of lines, planes, and subspaces, but it also encompasses more complex functions and transformations.
At the heart of linear algebra lie concepts such as matrices, determinants, and systems of linear equations. It provides the theoretical framework needed to understand methods for solving these systems.
For instance, the Wronskian is a concept that leverages linear algebra to determine whether a set of functions is linearly independent. It utilizes the principles of matrix operations and determinants to come to this conclusion.
Linear algebra therefore not only provides the tools needed to tackle the Wronskian problems, but it also gives us a language and framework to discuss transformations and relationships in multidimensional space. Understanding these relationships can be critical in advanced mathematics and various real-world applications such as in engineering and physics.