Question

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. \(x^{2}\left(1+x^{2}\right) y^{\prime \prime}-x\left(1-2 x^{2}\right) y^{\prime}+y=0\)

Short answer

Question: Find a fundamental set of Frobenius solutions for the given differential equation: $$x^2(1+x^2)y^{\prime\prime}+(1-2x^2)xy^{\prime}+x^4y=0$$ Answer: Two fundamental Frobenius solutions for the given differential equation are: Solution 1 (even coefficients): $$y_1(x) = \sum_{n=0}^\infty a_{2n} x^{2n+r}$$ Solution 2 (odd coefficients): $$y_2(x) = \sum_{n=0}^\infty a_{2n+1} x^{2n+1+r}$$ where the coefficients are determined by the recurrence relations: Even coefficients: $$a_{m+2} = \frac{(m+2r-1)(m+r)}{(m+2)(m+2r+1)} a_m$$ Odd coefficients: $$a_{m+2} = \frac{(m+2r)(m+r)}{(m+2)(m+2r+1)} a_m$$

Step-by-step solution

Convert to Standard Form

Divide both sides of the equation by x^2(1 + x^2): $$\frac{y^{\prime\prime}}{1+x^2} -\frac{(1-2x^2)}{x(1+x^2)}y^{\prime} +\frac{1}{x^2} y=0$$

Apply Frobenius Method

Assume a solution in the form of a power series: $$y(x) = \sum_{n=0}^\infty a_n x^{n+r}$$ Here, r is a constant and a_n are the coefficients. Differentiate y(x) with respect to x twice: $$y'(x) = \sum_{n=0}^\infty a_n(n+r)x^{n+r-1}$$ $$y''(x) = \sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-2}$$

Derive Recurrence Relations

Substitute y(x), y'(x), and y''(x) into the standard differential equation and equate like powers of x: $$\frac{\sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-2}}{1+x^2} -\frac{(1-2x^2)}{x(1+x^2)}\cdot\frac{\sum_{n=0}^\infty a_n(n+r)x^{n+r-1}}{x}+\frac{1}{x^2}\cdot\frac{ \sum_{n=0}^\infty a_n x^{n+r}}{x^2}=0$$ After simplifying, we get: $$\sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-2}-(1-2x^2)\sum_{n=0}^\infty a_n(n+r)x^{n+r-1}+ \sum_{n=0}^\infty a_n x^{n+r-2}=0$$ This will lead to a recurrence relation for the coefficients a_n.

Find Explicit Formulas for Coefficients

After equating coefficients of like powers of x, we derive the recurrence relation: $$a_{n+2} = \frac{(2n+2r-1)(n+r)}{(n+2)(n+2r+1)} a_n$$ From the recurrence relation, we find explicit formulas for the coefficients a_n for even and odd indices separately. For the even coefficients, let m = 2n and for the odd coefficients, let m = 2n + 1. Even coefficients: $$a_{m+2} = \frac{(m+2r-1)(m+r)}{(m+2)(m+2r+1)} a_m$$ Odd coefficients: $$a_{m+2} = \frac{(m+2r)(m+r)}{(m+2)(m+2r+1)} a_m$$

Write the Solutions

We found two solutions, one with even coefficients and one with odd coefficients. Write the two fundamental solutions with their corresponding coefficients: Solution 1 (even coefficients): $$y_1(x) = \sum_{n=0}^\infty a_{2n} x^{2n+r}$$ Solution 2 (odd coefficients): $$y_2(x) = \sum_{n=0}^\infty a_{2n+1} x^{2n+1+r}$$ These two solutions form a fundamental set of Frobenius solutions.

Key concepts

Fundamental Set of Solutions

In the realm of differential equations, finding a 'fundamental set of solutions' is akin to uncovering the building blocks for the general solution to an equation. When we talk about an ordinary second-order differential equation, two linearly independent solutions are required to form this set. These are crucial because any solution to the differential equation can be written as a combination of these fundamental solutions, reflecting the variety of initial conditions that might be applied to an equation.

In the context of the Frobenius method, the fundamental set is comprised of power series solutions that arise from different choices of the root 'r' in the assumed series solution. The coefficients of these series solutions are not arbitrary; they are determined through a specific process which ensures their independence and completeness as a set.

Recurrence Relation

A 'recurrence relation' is a sequence defined using preceding terms. In the case of differential equations solved by the Frobenius method, the recurrence relation provides a way to calculate the coefficients of the power series solution by relating each coefficient to previous ones. It serves as a repetitive formula that can be invoked to find all the coefficients of the series after establishing a few initial terms.

Dealing with the exercise at hand, we employ the Frobenius method to derive such a relation for the coefficients of our power series. This relation is tailored to reflect the structure of the differential equation, and it ensures consistency among all the coefficients, which ultimately shapes the form of the solutions.

Power Series Solutions of Differential Equations

Power series solutions are a powerful tool in solving differential equations, especially when the equations involve non-constant coefficients that complicate or even preclude a simple closed-form solution. To find a power series solution for an ordinary differential equation means expressing the solution as an infinite sum of powers of the independent variable, with each term having a coefficient that needs to be determined.

The Frobenius method is a specific technique that extends the power series method to handle singular points where the standard methods fail. It assumes a solution in the form of a power series starting with a term raised to the power of 'r', an unknown that can usually be determined by a characteristic equation. This approach transforms the differential equation into a recurrence relation for the coefficients of the series, which can then be computed to obtain the power series solution.

Ordinary Differential Equations

The mathematical expressions known as 'ordinary differential equations' (ODEs) relate functions to their derivatives, providing a way to model the change of a system. They are called ordinary to distinguish them from partial differential equations, which involve partial derivatives of multivariable functions. Many physical laws and phenomena can be described using ODEs, making them a fundamental concept in physics, engineering, and beyond.

An ODE involving power series solutions often comes into play when dealing with physical problems with variable coefficients or in the vicinity of singular points. Understanding the behavior of such an ODE is critical for grasping the dynamic nature of systems it represents. Each solution to an ODE represents a possible behavior of the system under different initial conditions, reflecting the diversity of real-world scenarios that the equation can model.