Question

Is the function \(2 x^{2}-1\) an eigenfunction of the operator \(-\left(3 / 2-x^{2}\right)\left(d^{2} / d x^{2}\right)+2 x(d / d x) ?\) If so, what is the eigenvalue?

Short answer

The function \(2x^2 - 1\) is an eigenfunction of the given operator \(-\left( \frac{3}{2} - x^2 \right) \frac{d^2}{dx^2} + 2x \frac{d}{dx}\) with the eigenvalue \(\lambda = 6\).

Step-by-step solution

Compute the first derivative of the function

Let's find \(f'(x)\), the first derivative of the function \(f(x)\). Using the power rule for derivatives, we have: \[f'(x) = \frac{d(2x^2 - 1)}{dx} = 4x\]

Compute the second derivative of the function

Next, let's find \(f''(x)\), the second derivative of the function \(f(x)\), using the first derivative we found. \[f''(x) = \frac{d(4x)}{dx} = 4\]

Apply the Operator to the Function

Now, we apply the given operator to the function using the first and second derivatives we computed. \[\hat{O}(f(x)) = -\left( \frac{3}{2} - x^2 \right) f''(x) + 2x f'(x)\] Plug in the derivatives we found for \(f'(x)\) and \(f''(x)\): \[\hat{O}(f(x)) = -\left( \frac{3}{2} - x^2 \right)(4) + 2x (4x)\]

Simplify and Check for Eigenvalue Equation

Finally, we simplify the result and check whether the eigenvalue equation holds: \[\hat{O}(f(x)) = -6 + 4x^2 + 8x^2 = 12x^2 - 6\] We see that this result is equal to 6 times the original function \(f(x)\): \[12x^2 - 6 = 6(2x^2 - 1)\] So, the Eigenvalue equation holds: \[\hat{O}(f(x)) = \lambda \cdot f(x)\] with \(\lambda = 6\), which is the eigenvalue we were looking for. #Conclusion# The function \(2x^2 - 1\) is indeed an eigenfunction of the given operator \(-\left( \frac{3}{2} - x^2 \right) \frac{d^2}{dx^2} + 2x \frac{d}{dx}\) with the eigenvalue \(\lambda = 6\).

Key concepts

Differential Operators

Differential operators are fundamental tools in calculus that involve derivatives or rates of change. They are used to express the change in a function with respect to one or more variables. In general, a differential operator contains derivatives like \( \frac{d}{dx} \) or \( \frac{d^2}{dx^2} \), which denote the first and second derivatives, respectively.

By applying a differential operator to a function, we transform it, often into a simpler or more insightful form. This transformation can tell us how the function behaves, such as its slope or concavity at any given point. In mathematical operations, differential operators help in forming equations that describe physical systems and processes.

Some common characteristics of differential operators include:

• Linear transformations when applied to functions.
• Possessing a characteristic polynomial when observed in matrix form.
• Used extensively in solving differential equations and modeling natural phenomena.

Being familiar with differential operators is essential for tackling advanced problems in calculus and physics.

Eigenvalue Problems

Eigenvalue problems are a major topic in linear algebra and calculus. They involve finding the eigenvalues and eigenfunctions of a system. In the context of differential equations, these problems help identify characteristic values that provide significant information about the system's behavior.

In an eigenvalue problem, you work with an operator \( \hat{O} \), which, when applied to a function \( f(x) \), returns the function times a scalar, known as the eigenvalue, \( \lambda \). This relationship is represented by the equation:

\[ \hat{O}(f(x)) = \lambda f(x) \]

This equation shows that \( f(x) \) remains proportional under the operation of \( \hat{O} \), with \( \lambda \) being the factor of proportionality.

In practical terms:

• Eigenvalues can show how systems like vibrations in a string or the modes of oscillation in a building change or persist over time.
• Each eigenvalue corresponds to an eigenfunction that represents a state or mode of the system.
• Analyzing these can simplify complex systems and reveal underlying patterns or resonances.

Understanding how to solve eigenvalue problems is critical for fields involving wave mechanics, quantum mechanics, and other areas where dynamic systems are analyzed.

Second Derivatives

The second derivative of a function provides valuable insights into its concavity and points of inflection. It is the derivative of a function's first derivative and can be denoted as \( f''(x) \) or \( \frac{d^2f}{dx^2} \).

When we compute the second derivative, we are essentially examining how the rate of change (the first derivative) itself changes. This allows us to explore more detailed aspects of a function's shape and behavior.

Key features of the second derivative include:

• If \( f''(x) > 0 \), the function is locally concave up (shaped like a cup), and the curve is bending upwards.
• If \( f''(x) < 0 \), the function is concave down (shaped like a cap), and the curve bends downwards.
• At points where \( f''(x) = 0 \), the function may have an inflection point, where the concavity changes.

By understanding second derivatives, students can predict and model more dynamic and complex behavior in not only mathematical functions, but in real-world applications, where anticipating changes is crucial for accurate analyses.