Question

Find the Wronskian \(W\) of a set of four solutions of $$ y^{(4)}+(\tan x) y^{\prime \prime \prime}+x^{2} y^{\prime \prime}+2 x y=0 $$ given that \(W(\pi / 4)=K\).

Short answer

Short Answer: To find the constant K, we must first define the Wronskian for a set of four solutions to the given differential equation. Next, we use the differential equation to find an expression for the fourth derivative term, which we then substitute into the Wronskian expression. Finally, we evaluate the Wronskian expression at x = π/4 to determine the value of K. However, since we do not have the specific solutions in this problem, we cannot find an explicit value for K.

Step-by-step solution

Define the Wronskian

For four solutions, let's say \(y_1(x)\), \(y_2(x)\), \(y_3(x)\), and \(y_4(x)\), the Wronskian, denoted by \(W\), is given by the determinant of the following square matrix: $$ W(y_1, y_2, y_3, y_4) = \begin{vmatrix} y_1 & y_2 & y_3 & y_4\\ y_1' & y_2' & y_3' & y_4'\\ y_1'' & y_2'' & y_3'' & y_4''\\ y_1''' & y_2''' & y_3''' & y_4''' \end{vmatrix} $$

Use the given differential equation

To simplify the Wronskian expression, we need to use the given differential equation: $$ y^{(4)}+(\tan x) y^{\prime \prime \prime}+x^{2} y^{\prime \prime}+2 x y=0 $$ Since each of \(y_i(x)\) (\(i = 1, 2, 3, 4\)) is a solution to this equation, it satisfies the given equation. Let's isolate the fourth derivative term (\(y_i^{(4)}\)) in the equation: $$ y_i^{(4)}=-(\tan x) y_i^{\prime \prime \prime}-x^{2} y_i^{\prime \prime}-2 x y_i $$

Replace the fourth derivative term in the Wronskian

Now, we will replace the fourth derivative term (\(y_i^{(4)}\)) in the Wronskian expression, using the isolated expression we found in Step 2: $$ W(y_1, y_2, y_3, y_4) = \begin{vmatrix} y_1 & y_2 & y_3 & y_4\\ y_1' & y_2' & y_3' & y_4'\\ y_1'' & y_2'' & y_3'' & y_4''\\ -(\tan x) y_1^{\prime \prime \prime}-x^{2} y_1^{\prime \prime}-2 x y_1 & -(\tan x) y_2^{\prime \prime \prime}-x^{2} y_2^{\prime \prime}-2 x y_2 & -(\tan x) y_3^{\prime \prime \prime}-x^{2} y_3^{\prime \prime}-2 x y_3 & -(\tan x) y_4^{\prime \prime \prime}-x^{2} y_4^{\prime \prime}-2 x y_4 \end{vmatrix} $$

Calculate the Wronskian at x = π/4

We are given that \(W(\pi / 4) = K\). Therefore, $$ K = W(y_1, y_2, y_3, y_4) \Bigg\rvert_{x=\frac{\pi}{4}} $$ Now, we can determine the constant K by substituting x = π/4 in the above expression. Note that this will require knowing the specific functions \(y_1\), \(y_2\), \(y_3\), and \(y_4\). In this problem, we are not given the specific solutions, so we cannot find an explicit value for the constant K. However, understanding how to find the Wronskian, use the given information, and evaluate it at a given point are crucial steps if the specific solutions were to be provided.

Key concepts

Determinants

The concept of a determinant is central to understanding the Wronskian. In mathematics, a determinant is a special scalar value that can be calculated from a square matrix. It tells us a lot about the properties of the matrix, such as whether it is invertible. For a 4x4 matrix like in our Wronskian problem, the determinant serves a key role with linear combinations of the rows or even columns.
To compute the determinant of a matrix, we systematically eliminate variables to reduce the matrix to simpler forms. In the Wronskian determinant:

• Each row of the matrix represents derivatives of the solutions.
• The order changes, going from the function itself, through its first, second, and third derivatives.
• If the determinant equals zero, the functions are not linearly independent.

Knowing how to calculate determinants is important as it influences whether solutions to a differential equation can form a complete and distinct solution set.

Linear Independence

Linear independence is a crucial concept when dealing with solution sets of differential equations. It involves determining whether a set of functions are independent of one another, meaning no function can be written as a linear combination of the others.
For instance, in our context with the Wronskian determinant, a non-zero determinant assures us of linear independence among solutions.

• If the Wronskian is zero, it suggests that at least two of the solutions are linearly dependent.
• On the other hand, a nonzero Wronskian at any point implies that the set of functions is linearly independent over the interval considered.

This attribute of functions is necessary for them to form a basis for the solution space of a differential equation, allowing us to construct general solutions by combining the base solutions in unique ways.

Higher-Order Differential Equations

Higher-order differential equations go beyond first and second order, involving derivatives of a function to higher degrees. In our problem, the differential equation involves a fourth derivative, making it a fourth-order equation.
Understanding these requires a repertoire of techniques given the complexity:

• They depict systems with more complicated behaviors, such as oscillations and reversals, demanding intricate solutions.
• The general solution involves several arbitrary constants, equivalent to the order of the highest derivative present.
• Higher-order equations are often decomposed into systems of first-order equations for easier analysis.

The ability to solve such equations is fundamental in fields like physics and engineering where they model real-world situations accurately.

Differential Equation Solutions

Solutions to differential equations represent functions that satisfy the equation. In context, they are functions like our hypothetical solutions \( y_1, y_2, y_3, \) and \( y_4 \).
The nature of these solutions influences the analysis and application:

• Exact solutions provide a definitive function satisfying the differential equation under given conditions.
• Approximate solutions, sometimes derived through series expansions or numerical methods, offer practical outcomes when exact forms are elusive or complex.
• General solutions aggregate all particular solutions, often including arbitrary constants found through integrating.

To apply solutions effectively, particularly in Wronskian or other analyses, understanding their differentiability, continuity, and behavior at certain points is important. This insight helps in evaluating or tweaking system parameters drastically.