Question

find the Wronskian of the given pair of functions. $$ x, \quad x e^{x} $$

Short answer

Answer: The Wronskian of the pair of functions x and xe^x is W(x) = x^2e^x.

Step-by-step solution

Write down the given functions

The given functions are: $$ f(x) = x \quad \text{and} \quad g(x) = xe^x $$

Compute derivatives of the given functions

Now, find the derivatives of both given functions with respect to x using basic differentiation rules: $$ f'(x) = \frac{d}{dx}(x) = 1 $$ and $$ g'(x) = \frac{d}{dx}(xe^x) = e^x + xe^x $$

Set up the Wronskian determinant

Now let's set up the Wronskian determinant W(x) for the pair of functions f(x) and g(x). The Wronskian determinant is given by: $$ W(x) = \begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix} $$ Substitute the functions and their derivatives we found in Steps 1 and 2: $$ W(x) = \begin{vmatrix} x & xe^x \\ 1 & e^x + xe^x \end{vmatrix} $$

Calculate the determinant

Now, calculate the determinant of the 2x2 matrix: $$ W(x) = x(e^x + xe^x) - xe^x $$

Simplify the expression

Finally, simplify the Wronskian expression: $$ W(x) = x^2e^x $$ So, the Wronskian of the given pair of functions x and xe^x is: $$ W(x) = x^2e^x $$

Key concepts

Differential Equations

Differential equations are equations that relate a function to its derivatives. They play a critical role in modeling real-life phenomena in physics, engineering, and economics. When you solve a differential equation, you are often finding a function or a set of functions that satisfy a given relationship involving derivatives.

Key aspects of differential equations include:

• Order: The highest derivative present in the equation. For example, if the highest derivative is the second derivative, the equation is a second-order differential equation.
• Linearity: Equations can be linear or nonlinear. Linear means that the function and its derivatives appear to the power of one. Nonlinear equations involve powers greater than one or functions of the derivatives.
• Solutions: Solutions to differential equations can be general (involving arbitrary constants) or particular (specific to certain conditions).

Working with differential equations often involves techniques such as separation of variables, integrating factors, and the use of the Wronskian to determine linear independence of functions.

Determinants

Determinants are important numerical values associated with square matrices, crucial for solving systems of linear equations and understanding linear transformations. The determinant of a matrix offers insights into the volume distortion caused by the corresponding linear transformation, and it indicates if a matrix is invertible.

For a 2x2 matrix, the determinant is calculated as follows:

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

In the context of finding a Wronskian, setting up a determinant helps signify the linear independence of solutions to differential equations. Calculating and simplifying determinants are key steps in understanding the relationships between functions.

Differentiation

Differentiation is a fundamental concept in calculus, involving the process of finding a derivative, which indicates the rate of change of a function. It transforms a function into another function that gives the slope of the original function at any given point.

Key differentiation rules include:

• Power Rule: For a function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
• Product Rule: If two functions \( u(x) \) and \( v(x) \) are multiplied, the derivative is \( (uv)' = u'v + uv' \).
• Chain Rule: If a function is composed of another function, the derivative is the derivative of the outer function times the derivative of the inner function.

In solving the Wronskian problem, differentiation is used to find the derivatives of the given functions, an essential step before setting up the Wronskian determinant.

Linear Independence

Linear independence is a concept in linear algebra where functions or vectors are considered independent if no function or vector in the set can be written as a linear combination of the others. In the context of differential equations, linear independence determines whether a set of solutions is unique and spans the solution space.

To test linear independence between functions, especially solutions of differential equations, the Wronskian provides a handy tool and is commonly used for two functions.

• Calculate the Wronskian of the functions. If the Wronskian is non-zero at at least one point in the interval of interest, the functions are linearly independent.

This property is crucial because it ensures that all important behaviors and solutions of the differential equation are captured effectively, without redundancy.