Problem 70 - Calculus Single Variable

Question

Involve the factorial numbers \(n !\), which were introduced in Section \(1.4 .\) They can be defined by \(0 !=1\) and, for a positive integer \(n, n !=n(n-1) \ldots 3 \ldots 2 \cdot 1\). In general, the Legendre polynomial \(P_{n}(x)\) (as defined in the preceding exercise) satisfies the differential equation $$\left(1-x^{2}\right) P_{n}^{\prime \prime}(x)-2 x P_{n}^{\prime}(x)+n(n+1) P_{n}(x)=0 .$$ Verify this equation for \(P_{5}(x)=\left(63 x^{5}-70 x^{3}+15 x\right) / 8\).

Step-by-Step Solution

Answer

The polynomial \( P_5(x) = \frac{63x^5 - 70x^3 + 15x}{8} \) satisfies the differential equation.

Step 1: Verify given polynomial

First, confirm that the given Legendre polynomial is correct for \( P_5(x) \). According to the problem, \( P_5(x) = \frac{63x^5 - 70x^3 + 15x}{8} \). This is the polynomial we will verify.

Step 2: Compute first derivative of P_5(x)

Differentiate the polynomial \( P_5(x) = \frac{63x^5 - 70x^3 + 15x}{8} \) to find \( P_5'(x) \). Use the power rule for differentiation:\[ P_5'(x) = \frac{1}{8} \left( 315x^4 - 210x^2 + 15 \right) \]

Step 3: Compute second derivative of P_5(x)

Differentiate \( P_5'(x) \) to find \( P_5''(x) \):\[ P_5''(x) = \frac{1}{8} \left( 1260x^3 - 420x \right) \]

Step 4: Substitute derivatives and polynomial into differential equation

Substitute \( P_5(x) \), \( P_5'(x) \), and \( P_5''(x) \) into the Legendre differential equation:\[ (1 - x^2) P_5''(x) - 2x P_5'(x) + n(n+1) P_5(x) = 0 \]where \( n = 5 \), so \( n(n+1) = 30 \).

Step 5: Simplify and verify the left side of the equation equals zero

Substitute the computed expressions:1.\[ (1 - x^2) \left( \frac{1}{8} \left( 1260x^3 - 420x \right) \right) = \frac{1}{8} \left( 1260x^3 - 420x - 1260x^5 + 420x^3 \right) \]2.\[ -2x \left( \frac{1}{8} (315x^4 - 210x^2 + 15) \right) = \frac{-2x}{8} \left( 315x^4 - 210x^2 + 15 \right) \]3.\[ n(n+1) P_5(x) = 30 \times \frac{63x^5 - 70x^3 + 15x}{8} \]Combine and simplify the terms to show they equal zero.

Key Concepts

Differential Equation

Differential equations are equations that involve an unknown function and its derivatives. In this particular context, the function we are looking at is a Legendre polynomial denoted by \( P_{n}(x) \). Each Legendre polynomial satisfies a specific differential equation, which acts as a rule. Essentially, the differential equation for the Legendre polynomial is expressed as: \[ \left(1-x^2\right) P_{n}''(x) - 2x P_{n}'(x) + n(n+1) P_{n}(x) = 0. \]This equation involves both the first and second derivatives of the polynomial. Derivatives are ways to show how a function changes at any point. The purpose of verifying a polynomial like \( P_5(x) \) with this differential equation is to confirm that the polynomial behaves according to this set rule. Breaking it down:

• \((1-x^2) P_{n}''(x)\): Involves multiplying the second derivative by \(1-x^2\).
• \(-2x P_{n}'(x)\): Involves multiplying the first derivative by \(-2x\).
• \(n(n+1) P_{n}(x)\): Simply multiply the original function by \(n(n+1)\), where \(n\) is the degree of the Legendre polynomial.

These components come together to provide a standardized form each Legendre polynomial must satisfy. Thus, the verification involves substituting the polynomial and its derivatives into this equation and showing all terms sum to zero, implying the differential equation is satisfied.

Factorial Numbers

Factorials are a simple yet powerful mathematical construct. They are denoted by an exclamation mark (\(!\)), and they represent the product of all positive integers up to a certain number. For instance, the factorial of 5, written as \(5!\), calculates as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \] Factorials play an important role in many areas of mathematics, including calculations involving permutations, combinations, and in this case, aiding in the formulation and behavior analysis of Legendre polynomials. When looking at Legendre polynomials, particularly within the differential equation:\[ n(n+1) \]The term \(n(n+1)\) originates from the properties of factorial numbers, helping to manage the polynomial's behavior with respect to its degree \(n\).Their critical role is to provide the necessary scaling that aligns with conventions in polynomial general solutions, furnishing the polynomial with the correct integer coefficients needed to yield consistent patterns within problem contexts. This often helps in ensuring that the computed solutions remain in harmony with expected outcomes of specific scenarios where polynomials are applicable.

Polynomial Verification

Verifying a polynomial involves confirming that it satisfies certain properties or equations, in this case, a differential equation. If we have a specific polynomial, such as \(P_5(x)\), verification is the process of substitution and simplification that ensures it fits a designated rule. With the given Legendre polynomial, \(P_5(x) = \frac{63x^5 - 70x^3 + 15x}{8}\), here is how verification proceeds:1. **Compute the Derivatives:** - **First Derivative:** Find \(P_5'(x)\). Differentiating gives the rate of change of the polynomial. - **Second Derivative:** Derive again to find \(P_5''(x)\). This derivative gives the change of the rate of change.2. **Substitute into the Differential Equation:** - You set up the defined components of the equation based on \((1-x^2)P''(x)\), \(-2xP'(x)\), and \(n(n+1)P(x)\). - Substitute \(P_5(x)\), \(P_5'(x)\), and \(P_5''(x)\) into the differential equation and make necessary simplifications.3. **Confirm the Equation Equals Zero:** - Once simplified, all terms should collectively sum to zero across all \(x\). If successful, this shows the polynomial meets the condition prescribed by the differential equation.This process of verification is crucial for validating the integrity and applicability of mathematical expressions to real-world scenarios. It takes logical steps, first achieving the necessary computations, and then ensuring conformity with broader mathematical rules.