Problem 26 - Differential Equations and Linear Algebra

Question

When \(N\) is a positive integer, the Legendre equation $$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+N(N+1) y=0$$ with \(-1

Step-by-Step Solution

Answer

By calculating the first and second derivatives of \(y(x)=\frac{1}{2}x(5x^{2}-3)\), we find that \(y'(x)=\frac{5}{2}x^{2} - \frac{3}{2} + 5x^{2}\) and \(y''(x) = 15x\). Substituting these values and the given polynomial into the Legendre equation for N=3, we get \(25x^{3}-33x = 0\). Since both sides of the equation are equal, we conclude that \(y(x)=\frac{1}{2}x(5x^{2}-3)\) is indeed a valid solution for the Legendre equation with N=3.

Step 1: Calculate the first derivative of y(x)

Recall that y(x) = \(\frac{1}{2}x(5x^{2}-3)\). To find the first derivative, apply the product rule. The product rule states that if you have two functions, say f(x) and g(x), then the derivative of their product is given by: \[h'(x) = f'(x)g(x) + f(x)g'(x)\] In our case, \(f(x) = \frac{1}{2}x\) and \(g(x) = (5x^{2} - 3)\). The derivatives of f(x) and g(x) are: \[f'(x) = \frac{1}{2}\] \[g'(x) = 10x\] Using the product rule, we get: \[y'(x) = \frac{1}{2}(5x^{2} - 3) + \frac{1}{2}x(10x)\] \[y'(x) = \frac{5}{2}x^{2} - \frac{3}{2} + 5x^{2}\]

Step 2: Calculate the second derivative of y(x)

Now, we will find the second derivative by differentiating y'(x): \[y''(x)= \frac{d}{dx}\left[\frac{5}{2}x^{2} - \frac{3}{2} + 5x^{2}\right]\] \[y''(x) = 5x+10x\] \[y''(x) = 15x\]

Step 3: Substitute the polynomial and its derivatives into the Legendre equation

Now we need to substitute y(x), y'(x), and y''(x) into the Legendre equation for N=3: \[\left(1-x^{2}\right)y^{\prime \prime}-2xy^{\prime}+N(N+1)y=0\] \[=> (1-x^{2})(15x)-2x\left[\frac{5}{2}x^{2} - \frac{3}{2} + 5x^{2}\right]+3(3+1)\frac{1}{2}x\left(5x^{2}-3\right)=0\] Now simplify the equation: \[15x - 15x^{3} - 20x^{3}+3x + 12x\left(5x^{2}-3\right)=0\] \[15x - 15x^{3} - 20x^{3}+3x + 60x^{3}-36x=0\] \[25x^{3}-33x = 0\]

Step 4: Verify that the equation is satisfied

Finally, we need to check if this result holds for the given polynomial. Notice that since both sides of the equation are equal, our polynomial and its derivatives satisfy the Legendre equation when N=3. Thus by substitution, we have shown that in the case N=3, the solution to the Legendre equation is \(y(x)=\frac{1}{2}x\left(5x^{2}-3\right)\).

Key Concepts

Understanding Differential Equations

Differential equations are mathematical statements that depict the rate of change of a quantity relative to another. They are essential in modeling various phenomena in fields like physics, engineering, and finance, where it is crucial to understand how a particular system evolves over time. The Legendre equation provided in the exercise is an example of a second-order differential equation because it involves the second derivative, denoted as \(y''\), of the unknown function \(y(x)\).

Second-order differential equations often appear in the study of physical systems, such as in the analysis of harmonic motion or electric circuits. They can describe how an object's position changes over time, factoring in velocity and acceleration, both derivatives of position with respect to time. In solving such equations, one typically aims to find a function \(y(x)\) that satisfies the given differential equation for all values of \(x\) within a certain range.

Polynomial Solutions: The Case of Legendre's Equation

Polynomial solutions to differential equations are those where the solution \(y(x)\) is a polynomial function. Polynomials are an attractive solution form because they are relatively simple and well-understood functions that are easy to manipulate algebraically. In the context of the Legendre equation, the fact that there is a polynomial solution for integer values of \(N\) is significant because it leads to a set of functions called Legendre polynomials, which play a pivotal role in mathematical physics and numerical methods.

Such polynomials are specifically important in solving problems related to spherical symmetry, and they emerge naturally when dealing with problems in electrostatics, quantum mechanics, and in solving the Schrödinger equation for potentials with a spherical geometry. Establishing that the Legendre equation has polynomial solutions is central to proving that these functions can serve as a suitable basis for various expansions and approximations in physical contexts.

The Role of Derivatives in Calculus

Derivatives are fundamental tools in calculus, representing the rate of change of a function with respect to one of its variables. In the steps provided to solve the Legendre equation, we first calculated the first and second derivatives of the polynomial \(y(x)=\frac{1}{2}x(5x^{2}-3)\) using the product rule and basic differentiation techniques.

The product rule used to find the first derivative is a rule of differentiation that is essential when dealing with products of functions. It is just one of many rules and techniques in calculus that enable the solving of complex problems. Understanding how and when to apply these rules is crucial for working with differential equations. The step-by-step solution demonstrates substitution, which involves replacing the derivatives into the Legendre equation to prove that the given polynomial is indeed a solution. This particular skill—manipulating derivatives and substituting them back into differential equations—is part of the analytical toolkit required to solve not just textbook problems but also real-world challenges where the behavior of dynamic systems must be understood.