Question

Use Theorem 7.2 .2 to show that the power series in \(x\) for the general solution of $$ \left(1+\alpha x^{2}\right) y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0 $$ is $$ y=a_{0} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2 j)\right] \frac{x^{2 m}}{(2 m) !}+a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2 j+1)\right] \frac{x^{2 m+1}}{(2 m+1) !} $$

Short answer

Question: Show that the power series in \(x\) for the general solution of the given differential equation is as stated using Theorem 7.2.2. Answer: To show that the power series in \(x\) for the general solution of a given differential equation is as stated using Theorem 7.2.2, we need to follow these steps: 1. Recall Theorem 7.2.2 and write down the given differential equation in the standard form. 2. Identify the coefficients of the differential equation. 3. Compute the power series coefficients for the even and odd powers of \(x\). 4. Write down the general solution as a power series using the computed coefficients. Following these steps, we construct the general solution of the given differential equation as a power series in the form: $$y = a_{0} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2j)\right] \frac{x^{2m}}{(2m)!} + a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2j+1)\right] \frac{x^{2m+1}}{(2m+1)!}$$

Step-by-step solution

Recall Theorem 7.2.2 and write down the given differential equation in the standard form

Theorem 7.2.2 states that if the coefficients of a differential equation are analytic functions, then the general solution of the differential equation can be expressed as a power series. First, let's write down the given differential equation: \(\left(1+\alpha x^{2}\right) y^{\prime \prime}+\beta x y^{\prime}+\gamma y=0\)

Write down the coefficients of the equation

In order to apply Theorem 7.2.2, we must identify the coefficients of the given differential equation. They are as follows: \(p(x) = 1 + \alpha x^2\) \(q(x) = \beta x\) \(r(x) = \gamma\)

Compute the power series coefficients

As per Theorem 7.2.2 and the given formula in the exercise, let's compute the power series coefficients of the general solution of the differential equation: - For the even powers of \(x\) (\(x^{2m}\)): \((-1)^{m}\left[\prod_{j=0}^{m-1} p(2 j)\right] \frac{x^{2m}}{(2m)!}\) - For the odd powers of \(x\) (\(x^{2m+1}\)): \((-1)^{m}\left[\prod_{j=0}^{m-1} p(2 j+1)\right] \frac{x^{2m+1}}{(2m+1)!}\)

Write down the general solution as a power series

Now that we have computed the power series coefficients, we can write down the general solution of the differential equation as a power series in the form: $$y = a_{0} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2j)\right] \frac{x^{2m}}{(2m)!} + a_{1} \sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1} p(2j+1)\right] \frac{x^{2m+1}}{(2m+1)!}$$ This is the desired power series representation of the general solution of the given differential equation using Theorem 7.2.2.

Key concepts

Understanding Theorem 7.2.2

Theorem 7.2.2 is a pivotal concept in the study of differential equations, particularly when dealing with solutions that can be expressed as power series. The theorem states that if the coefficients of a differential equation are analytic functions, then the solution to the differential equation can be expressed as a power series around the point where the coefficients are analytic.

Analytic functions are those that can be locally expressed by a convergent power series. This is crucial because it ensures that we can apply operations such as differentiation and integration term-by-term to the series, allowing for straightforward manipulation and calculation of the series coefficients required for the general solution.

Finding the General Solution to Differential Equations

The general solution of a differential equation represents the full set of possible solutions that satisfy the equation, often encompassing an infinite number of particular solutions. It is characterized by containing arbitrary constants that accommodate various initial conditions or boundary values. It is important to establish the general solution because it provides a comprehensive understanding of the behavior of the differential system under study.

Using power series solutions is especially advantageous for equations with coefficients that are not simple functions or for which the solution is difficult to express in closed form. In such cases, power series allow us to approximate the solution to any desired degree of accuracy by considering enough terms of the series.

Analytic Functions in Differential Equations

In the realm of differential equations, an analytic function is one that can be expanded in a power series around a given point within its domain. The relation to differential equations comes into play when these functions serve as coefficients. The analytic nature of these coefficients simplifies the process of solving differential equations because it permits the construction of a power series solution. If the coefficients of an ordinary differential equation are analytic at a point, it implies that there is a power series solution convergent in a neighborhood of that point.

Working with analytic functions is essential in the context of Theorem 7.2.2, as it ensures that the processes used to find the series coefficients will lead to a valid solution convergent in a region around the point of expansion.

Power Series Representation

A power series representation of a function is an expansion that expresses the function as an infinite sum of terms, each of which is a constant multiplied by a power of the variable. In the context of differential equations, representing the solution as a power series enables us to handle even very complex differential equations by breaking them down into more manageable infinite sums.

Formulating a solution in terms of power series involves calculating the coefficients that accompany each term of the series. As indicated in the step-by-step exercise presented above, these coefficients are typically determined based on the structure of the differential equation and the nature of its coefficients. The elegance of the power series method lies in its ability to approximate the behavior of functions that may not have simple explicit solutions, offering a flexible approach to tackling a vast array of problems in differential equations.