Question

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(x)=e^{3 x}, \quad g(x)=e^{3(x-1)}\)

Short answer

Answer: Linearly dependent.

Step-by-step solution

Find the first derivatives of the given functions

First, we need to find the first derivatives for each function: \(f(x) = e^{3x}\) \(f'(x) = 3e^{3x}\) \(g(x) = e^{3(x-1)}\) \(g'(x) = 3e^{3(x-1)}\)

Calculate the Wronskian of the functions

The Wronskian for a pair of functions is defined as: \(W(x) = \det(\begin{bmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{bmatrix})\) We will calculate the Wronskian using the functions and their derivatives: \(W(x) = \det(\begin{bmatrix} e^{3x} & e^{3(x-1)} \\ 3e^{3x} & 3e^{3(x-1)} \end{bmatrix})\) To find the determinant of this matrix, we perform the following operations: \(W(x) = e^{3x} \cdot 3e^{3(x-1)} - e^{3(x-1)} \cdot 3e^{3x}\) Now we can eliminate the repeated term and simplify the equation: \(W(x) = 3e^{3x}e^{3(x-1)} - 3e^{3(x-1)}e^{3x} = 0\)

Analyze the result to determine linear independence or dependence

Since the Wronskian is equal to 0, it follows that the given pair of functions \(f(x)=e^{3x}\) and \(g(x)=e^{3(x-1)}\) are linearly dependent.

Key concepts

Differential Equations

Differential equations play a critical role in mathematics, acting as equations that describe the relationship between a function and its derivatives. They are widely used in various fields such as physics, engineering, and economics to model how things change over time. A differential equation can be simple, involving a single derivative, or complex, involving multiple derivatives and variables.
One of the key types of differential equations is the linear differential equation, where the unknown function and its derivatives appear to the first power and do not multiply each other. Understanding whether the functions forming the solutions to these equations are linearly independent or not is vital. This involves concepts such as the Wronskian determinant, which helps in determining solutions' characteristics.
Studying differential equations requires a solid understanding of calculus and how derivatives represent change. It provides a foundational framework for modeling dynamic systems and solving complex real-world problems.

Wronskian Determinant

The Wronskian determinant is a mathematical tool used to determine whether a set of functions are linearly independent. This can be useful, for example, in solving systems of differential equations where knowing the independence of solutions is crucial.
The Wronskian of two functions, say \(f(x)\) and \(g(x)\), is calculated by setting up a determinant of a matrix that includes the functions and their first derivatives:

• The first row consists of the functions themselves: \(f(x)\) and \(g(x)\).
• The second row includes their derivatives: \(f'(x)\) and \(g'(x)\).

The Wronskian is given by:\[W(x) = \det\left(\begin{array}{cc} f(x) & g(x) \ f'(x) & g'(x) \end{array}\right)\].
For example, if \(W(x) = 0\) at a particular point, it suggests that the functions may be linearly dependent in that interval. It is an excellent check for independence among solutions to differential equations, helping to ensure we have a complete set of solutions when addressing linear problems.

Linearly Dependent Functions

Linearly dependent functions are those that can be expressed as a linear combination of each other with non-zero coefficients. In simpler terms, if one function can be composed by multiplying another function by a constant, they are linearly dependent. This concept is crucial in understanding the behavior of solutions to differential equations.
For instance, considering the functions \(f(x) = e^{3x}\) and \(g(x) = e^{3(x-1)}\) from the Wronskian calculation, they were found to be linearly dependent because their Wronskian is zero. This means that one is just a shifted version of the other: \(g(x) = e^{-3}f(x)\).
Understanding this dependence helps in solving problems where the independence of solutions is necessary. In the context of a system of equations, if all solutions are linearly dependent, it might indicate redundancies or symmetries in the system. Hence, in mathematics, ensuring functions are independent when needed is fundamental to forming a solid solution basis.