Question

The Wronskian of two functions is \(W(t)=t \sin ^{2} t .\) Are the functions linearly independent or linearly dependent? Why?

Short answer

Question: Determine if the functions whose Wronskian is given by \(W(t) = t \sin^2 t\) are linearly independent or linearly dependent. Answer: The functions are linearly independent.

Step-by-step solution

Analyze the Wronskian of given functions

The Wronskian \(W(t)\) is given as \(t \sin^2 t\). We need to find the values of \(t\) for which the Wronskian is nonzero.

Check for Wronskian being nonzero

The given Wronskian is \(W(t) = t \sin^2t\). Let's analyze this function: 1. For \(t > 0\) and \(t \neq n\pi\) for integer n, we see that \(\sin^2t\) will be nonzero, and hence \(W(t)\) will be nonzero. 2. For \(t < 0\), \(t\) is negative and \(\sin^2t \geq 0\), but for \(t \neq n\pi\) for integer n, the product \(t\sin^2t\) will be negative and thus nonzero. 3. For \(t = n\pi\) for integer n, we get \(W(n\pi) = n\pi \sin^2(n\pi) = 0\).

Determine Linear Independence or Dependence

From step 2, we see that \(W(t)\) is nonzero for some values of \(t\). Therefore, the functions are linearly independent.

Key concepts

Differential Equations

Differential Equations are equations that involve an unknown function and its derivatives. They are a cornerstone of mathematical modeling, allowing us to describe natural phenomena like physics, engineering, economics, and more. Typically, they come in several forms like ordinary differential equations (ODEs), and partial differential equations (PDEs). Solving a differential equation means finding a function that satisfies the equation. Often, solutions to differential equations are unique only under certain conditions, which leads us to the concept of boundary value problems.

In practice, solving a differential equation analytically can be complex, and often, numerical methods are employed. These solutions not only provide explicit functions but also impart understanding of the behavior of the system described by the equation.

Linear Independence

Linear independence is a fundamental concept in mathematics, particularly in linear algebra and differential equations. A set of functions (or vectors) is said to be linearly independent if no function in the set can be written as a linear combination of the others. In other words, the only solution to the equation \(c_1f_1(t) + c_2f_2(t) + ... + c_nf_n(t) = 0\), for constants \(c_1, c_2, ..., c_n\), is when all the constants are zero.

Importance in Differential Equations

Linear independence amongst solutions to a differential equation is crucial. It is because the general solution is often expressed as a combination of linearly independent solutions. The uniqueness of solutions is partly dictated by this linear independence. Hence, having tools to test for linear independence, like the Wronskian, is key to solving differential equations effectively.

Wronskian Nonzero

The Wronskian is a determinant used as a test for linear independence of solutions to differential equations. Specifically, if the Wronskian of a set of functions is nonzero at some point within an interval, the functions are linearly independent on the interval. It is important to note the converse is not always true; a zero Wronskian does not necessarily imply linear dependence, unless the functions are analytic.

The Wronskian's practical approach relies on calculating the determinant of a matrix comprised of the functions and their derivatives. For example, in the case of two functions \( f \) and \( g \), their Wronskian is \( W(t) = f(t)g'(t) - f'(t)g(t) \). If this calculation results in a nonzero value, one can conclude that \( f \) and \( g \) are linearly independent.

Boundary Value Problem

A Boundary Value Problem (BVP) is a type of differential equation where the solution is determined based on conditions given at the boundaries of the domain, such as at the endpoints of an interval. For ordinary differential equations, these conditions typically specify the values of the solution or its derivatives at the boundaries.

Solving BVPs

BVPs are more challenging to solve than initial value problems because the solution often requires satisfying conditions at more than one point. Furthermore, the existence of a solution to a BVP can be greatly affected by the specifics of the boundary conditions. For example, the Sturm-Liouville problems, an important class of BVPs, arise in many physical applications like quantum mechanics and heat conduction. Solving a BVP can involve a mix of analytic techniques, such as separation of variables, and numerical methods when analytic solutions are intractable.