Question

Use the method of Problem 11 to transform the given equation into the form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\) $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0, \quad \text { Chebyshev equation } $$

Short answer

Question: Transform the given Chebyshev equation into the form \((p(x)y^{'})'+q(x)y=0\): Given equation: \(\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0\) Answer: The transformed equation is \([(1-x^{2})y''] + [(-2x-x)y'] + \alpha^{2}y = 0\), with \(p(x) = 1-x^2\) and \(q(x) = \alpha^2\).

Step-by-step solution

Finding \(p(x)\) and \(q(x)\)

In our given equation, \(p(x) = 1-x^2\) and \(q(x) = \alpha^2\). Now we need to find expressions for \(y'\) and \(y''\).

Differentiating \(p(x)y'\)

To find the expression for \((p(x) y^{'})'\), we need to differentiate \(p(x)y'\) with respect to \(x\): $$ \frac{d}{dx}(p(x)y') = \frac{d}{dx}((1-x^2)y') $$

Applying Product Rule of Differentiation

Using the product rule of differentiation, \(\frac{d}{dx}(fg) = f'g+fg'\), we differentiate \((1-x^2)y'\): $$ \frac{d}{dx}((1-x^2)y') = (-2x)y' + (1-x^2)y'' $$ Now, the given equation can be reformulated using the expression found in Step 3:

Rearrange the Equation

Substitute the expression from Step 3 into the Chebyshev equation: $$ (-2x)y' + (1-x^2)y'' - xy' + \alpha^2y = 0 $$ Now collect terms with \(y'\) and \(y\): $$ [(1-x^{2})y''] + [(-2x-x)y'] + \alpha^{2}y = 0 $$ We have now transformed the given Chebyshev equation into the desired form: $$ \left[p(x) y^{\prime}\right]'+q(x) y=0 $$ With \(p(x) = 1-x^2\), and \(q(x) = \alpha^2\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are crucial in modeling the behavior of various physical systems, such as the motion of celestial bodies, the flow of fluids, and electrical circuits.

At their core, differential equations express how a quantity changes as it depends on other quantities. For example, the rate at which a population grows may be dependent on its current size, or the speed of a car might depend on the force applied to it and the resistance it encounters.

The Chebyshev equation, \( (1-x^{2}) y^{\'\'} - x y^{\'\text{ }}+ \text{ }\alpha^{2} y=0 \) is a specific type of differential equation with significant applications in mathematical physics and engineering. In this context, the exercise focuses on transforming the Chebyshev equation into a more general form that is easier to analyze and solve.

Boundary Value Problems

Boundary value problems (BVPs) are a class of differential equations where the solution is determined not just by the equation itself but also by conditions, known as boundary conditions, that are applied to the boundaries of the domain in which the solution is sought.

These problems arise naturally in many practical contexts—thermal conduction, wave mechanics, and static potentials, to name a few. Solving a BVP involves determining a function that satisfies not only the differential equation but also meets predefined criteria at the boundaries of the domain.

The Chebyshev equation from our exercise could lead into such a boundary value problem, depending on the conditions applied at the endpoints of an interval. These conditions often reflect physical constraints or symmetry properties in the system being modeled.

Product Rule of Differentiation

The product rule of differentiation is a fundamental calculus technique used to find the derivative of the product of two functions. It can be expressed as \( (fg)^\text{'} = f^\text{'}g + fg^\text{'} \), where \( f \) and \( g \) are functions of \( x \) and \( f^\text{'} \) and \( g^\text{'} \) are their respective derivatives.

This rule is essential when dealing with compound expressions in differential equations, where a function and its derivative are multiplied together. In the case of the Chebyshev equation exercise, the rule is applied to differentiate the term \( (1-x^2)y^\text{'} \) properly, resulting in the correct transformation of the equation into a standard form from which solutions can be more readily obtained.