Question

In each of Problems 12 through 15 use the method of Problem 11 to transform the given equation into the form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\) $$ y^{\prime \prime}-2 x y^{\prime}+\lambda y=0, \quad \text { Hermite equation } $$

Short answer

The functions p(x) and q(x) are found as follows: $$p(x) = e^{-x^2}$$ $$q(x) = \lambda e^{-x^2}$$

Step-by-step solution

Identify the coefficient of the first derivative and zeroth derivative terms

From the Hermite equation, we can identify the coefficients of \(y^{\prime}\) and y as follows: $$ y^{\prime \prime}-2 x y^{\prime}+\lambda y=0 $$ Coefficient of \(y^{\prime}\): \(-2x\) Coefficient of y: \(\lambda\)

Find p(x) using the equation \(p(x) = e^{\int -2x dx}\)

To obtain the function p(x), we need to calculate the integral of -2x and apply the exponential function: $$ p(x) = e^{\int -2x dx} $$ Calculate the integral: $$ \int -2x dx = -x^2 + C $$ However, since we are only interested in the multiplicative function, we can disregard the constant C: $$ p(x) = e^{-x^2} $$

Multiply the Hermite equation by p(x)

Now, we need to multiply the entire Hermite equation by p(x): $$ e^{-x^2} \left(y^{\prime \prime} - 2xy^{\prime} + \lambda y \right) = 0 $$

Rewrite the equation as \([p(x) y^{\prime}]^{\prime} + q(x) y=0\)

We can now rewrite this equation as: $$ \left[e^{-x^2} y^{\prime}\right]^{\prime} + q(x) y = 0 $$ To find the function q(x), we will compare the coefficients of y in the last equation we derived. From the last equation, we can find q(x) as the coefficient of y term's multiplicative factor: $$ q(x) = \lambda e^{-x^2} $$ Now, we can rewrite the equation in the standard form: $$ \left[e^{-x^2} y^{\prime}\right]^{\prime} + \left(\lambda e^{-x^2}\right) y = 0 $$ This is the desired form of the given Hermite equation.

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are central in modeling the behavior of various systems in engineering, physics, economics, and other fields. For instance, they are used to describe the motion of particles, the growth of populations, or the cooling of objects over time.

When solving a differential equation, such as the Hermite equation given in the exercise, it's crucial to identify the order of the highest derivative, which, in this case, is the second derivative, denoted by \(y''\). The exercise asks to transform the equation into a specific form, which involves finding a multiplier \(p(x)\) that simplifies the equation into a recognizable standard format. This is useful because it can lead to finding solutions more systematically, allowing us to analyze the behavior of the system in more detail.

The step-by-step solution walks through how to determine the function \(p(x)\) by integrating a coefficient from the original equation. In this example, integral calculus is applied to find \(p(x)\), which will ultimately help in simplifying the original Hermite equation into a standard form that's easier to work with.

Sturm-Liouville Problem

The Sturm-Liouville problem is a particular type of differential equation that arises in the context of finding eigenfunctions and associated eigenvalues. It is named after French mathematicians Jacques Charles François Sturm and Joseph Liouville and pertains to second-order linear differential equations. The general form of a Sturm-Liouville problem is: \[\left[ p(x) y' \right]' + q(x) y = -\lambda w(x) y\] where \(p(x)\), \(q(x)\), and \(w(x)\) are known functions, and \(\lambda\) is the eigenvalue. The Hermite equation in the exercise resembles a Sturm-Liouville form which guides us to find corresponding eigenvalues and eigenfunctions.

The importance of the Sturm-Liouville theory is profound since it serves as a foundation for understanding various physical systems that can be modeled by eigenvalues problems, such as vibrations, heat conduction, and quantum mechanics. Moreover, Sturm-Liouville problems help in developing series solutions to complex differential equations, expanding functions into eigenfunction series, somewhat analogous to breaking down a signal into Fourier series.

Eigenvalue Problems

Eigenvalue problems are mathematical challenges where we look for solutions to equations involving differential operators. In these problems, the goal is to find the 'special numbers' known as eigenvalues, along with their corresponding eigenfunctions, which satisfy the equation. These concepts are not only central in mathematics but also in many branches of science and engineering.

The term 'eigen' comes from German, meaning 'own' or 'characteristic'. In the context of differential equations, finding the eigenvalues helps characterize the behavior of systems under study. For example, in the Hermite equation from the exercise, \(\lambda\) represents the eigenvalue, and finding it, along with its associated eigenfunction, is crucial for fully understanding and solving the Hermite differential equation.

Eigenvalue problems like the one presented often lead to a series of numbers (eigenvalues) and functions (eigenfunctions), which are very useful in the expansion of functions over different domains. They are also key in the study of linear transformations in linear algebra, determining pivotal attributes of physical systems like stability, resonant frequencies, and normal modes in dynamics.