Problem 116 - Calculus 3

Question

The differential equation \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) is a Bessel equation of order 1\. Use a power series of the form $$y=\sum_{n=0}^{\infty} a_{n} x^{n}$$ to find the solution.

Step-by-Step Solution

Answer

The solution is a combination of Bessel functions: \( y = c_1 J_1(x) + c_2 Y_1(x) \).

Step 1: Write the Power Series

Assume the solution is of the form \( y = \sum_{n=0}^{\infty} a_{n} x^{n} \). We also need the derivative forms: \( y' = \sum_{n=1}^{\infty} na_{n} x^{n-1} \) and \( y'' = \sum_{n=2}^{\infty} n(n-1)a_{n} x^{n-2} \).

Step 2: Substitute into the Differential Equation

Substitute the series and its derivatives into the differential equation:\[ x^2 \sum_{n=2}^{\infty} n(n-1)a_{n} x^{n-2} + x\sum_{n=1}^{\infty} na_{n} x^{n-1} + \sum_{n=0}^{\infty} a_{n}(x^{2}-1)x^{n} = 0 \].

Step 3: Simplify the Expression

Rewrite all terms so they have the same power of \( x \). This involves changing the indices of the sums to align the powers of \( x \):\[ \sum_{n=0}^{\infty} \left( n(n-1)a_{n} + na_{n} + (1 - x^2)a_n \right) x^n = 0 \].

Step 4: Solve the Recurrence Relation

Since the series equals zero for all \( x \), all coefficients must be zero. For non-trivial solutions, solve \( n(n-1)a_{n} + (1-x^2)a_n = 0 \). Simplifying gives a recurrence relation for \( a_n \).

Step 5: Find the First Few Terms

Assume initial conditions (e.g., \( a_0 = 1 \), \( a_1 = 0 \)). Use the recurrence relation to find \( a_2, a_3,.. \) and so forth, forming a series solution.

Key Concepts

Power Series Solution

When solving differential equations, it's often useful to express the solution as an infinite sum or power series. For Bessel's equation, we assume the solution can be written in the form of a power series: \( y = \sum_{n=0}^{\infty} a_{n} x^{n} \).
\( \)This expression is essentially a series expansion of the function \( y \) in terms of powers of \( x \).
Here's how it breaks down:

• \( a_n \) are the coefficients that determine the contribution of each power of \( x \) to the function.
• The series starts at \( n = 0 \) and goes to infinity, so it includes all powers of \( x \) from zero upwards.

By substituting this representation into the given differential equation, we're able to derive relationships between these coefficients, which ultimately allow us to determine the form of the solution. The power series is especially helpful because it can often provide a valid solution even where standard functions cannot. This method leverages the fundamental property of power series to represent functions in situations where traditional forms are difficult to apply.

Differential Equations

Differential equations like Bessel's equation involve functions and their derivatives.
These equations allow scientists and engineers to model real-world phenomena such as heat conduction, wave propagation, and quantum mechanics.The particular differential equation presented, \( x^{2} y^{\prime \prime} + x y^{\prime} + (x^{2} - 1) y = 0 \), is a second-order linear differential equation.

• Here, \( y^{\prime} \) denotes the first derivative of \( y \) with respect to \( x \), and \( y^{\prime\prime} \) denotes the second derivative.
• The coefficients of \( y^{\prime} \) and \( y^{\prime\prime} \) depend on \( x \), making it a point of interest as it resembles Bessel's equation, which commonly appears in physical applications involving cylindrical or spherical symmetry.

Differential equations are often solved using various techniques depending on their complexity and form. For Bessel's equation, the power series method is particularly apt since it allows for the systematic calculation of solutions when traditional algebraic manipulation is complex or infeasible.

Recurrence Relation

A crucial step in solving a differential equation via power series is deriving a recurrence relation. This relation helps you determine the series coefficients \( a_n \).
It arises from setting the entire power series expression equal to zero, as required by the differential equation.During the process:

• You align all terms in the series to have the equivalent powers of \( x \).
• Consequently, a condition forms that tells us how each coefficient relates to others.
• This results in a step-by-step process for finding each successive term from previous ones, known as a recurrence relation.

For the problem we're examining, this involves simplifying to the point where the expression involves sums of terms like \( n(n-1) a_{n} + na_{n} + (1 - x^2)a_n = 0 \). By comparing coefficients of like powers of \( x \), you develop a formula to compute each \( a_n \). Naturally, initial conditions or given values for \( a_0 \) or \( a_1 \) start this calculative journey, ensuring you get determinate values for the power series, thus forming a complete solution.