Question

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

Short answer

Question: Transform the given Bessel equation into the form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\) using the method of Problem 11. Given Bessel equation: \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y\) Solution: 1. Divide the given equation by \(x^2\): \(y^{\prime\prime} + \frac{1}{x}y^{\prime} + \left(1-\frac{v^2}{x^2}\right)y=0\) 2. Identify the integrating factor: \(\mu(x) = e^{\int p(x) dx} = e^{\ln(x)} = x\) 3. Multiply the equation by the integrating factor: \(x(y^{\prime\prime} + \frac{1}{x}y^{\prime} + \left(1-\frac{v^2}{x^2}\right)y)=0\) The final equation is in the form \(\left[p(x) y^{\prime}\right]'+q(x) y=0\) where \(p(x)=x\) and \(q(x)=x\left(1-\frac{v^2}{x^2}\right)\).

Step-by-step solution

Divide the given equation by \(x^2\)

Divide the Bessel equation by \(x^2\) to simplify: $$\frac{1}{x^2}(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right) y)=\frac{1}{x^2}(x^{2}y'')+ y'+\left(1-\frac{v^2}{x^2}\right) y=0$$ So we have: $$y^{\prime\prime} + \frac{1}{x}y^{\prime} + \left(1-\frac{v^2}{x^2}\right)y=0$$

Identify the integrating factor

According to the method of Problem 11, we need to find an integrating factor of the form \(\mu(x) = e^{\int p(x) dx}\) where \(p(x)\) can be taken from our simplified equation as \(\frac{1}{x}\). Now we compute the integral: $$\int p(x) dx = \int \frac{1}{x} dx = \ln(x) + C$$ For our purpose, we do not need to consider the constant \(C\). The integrating factor is then: $$\mu(x) = e^{\ln(x)} = x$$

Multiply the equation by the integrating factor

Now multiply our simplified equation by the integrating factor \(x\): $$x(y^{\prime\prime} + \frac{1}{x}y^{\prime} + \left(1-\frac{v^2}{x^2}\right)y)=0$$ Expand the equation: $$x y^{\prime\prime} + y^{\prime} + x\left(1-\frac{v^2}{x^2}\right)y=0$$ So we have achieved the desired form: $$\left[p(x) y^{\prime}\right]'+q(x) y=0$$ where \(p(x)=x\) and \(q(x)=x\left(1-\frac{v^2}{x^2}\right)\).

Key concepts

Differential Equations

Differential equations are powerful tools in mathematics used to describe various phenomena in physics, engineering, biology, and more. They consist of equations involving functions and their derivatives. In a differential equation, the unknown entity is a function rather than a number, which is the case in algebra. One such equation is the Bessel equation, characteristic of certain types of wave propagation and static potentials.

The Bessel equation takes on the form \( x^{2} y^{\backprime \backprime}+x y^{\backprime}+(x^{2}-v^{2}) y=0 \), where \( y^{\backprime} \) and \( y^{\backprime \backprime} \) represent the first and second derivatives of the function \( y \) with respect to \( x \) and \( v \) is a parameter that often arises in the context of boundary value problems. In this exercise, we transform the given Bessel equation into a more standard format that is easier to analyze and solve.

Integrating Factor Method

The integrating factor method is a critical technique used to solve certain types of differential equations, especially those that are not readily separable. The method involves finding a multiplier, known as an integrating factor, which when multiplied by the differential equation, simplifies it into a form that can be integrated directly.

For the Bessel equation, the integrating factor is found by identifying the function \( p(x) \) that we would integrate to get the integrating factor \( \( \)mu(x) = e^{\int p(x) dx} \). In the given exercise, \( p(x) \) turns out to be \( \frac{1}{x} \) leading us to an integrating factor \( \)mu(x) = x \. We then use this factor to convert the Bessel equation into a more workable form, essentially priming it for further analysis and eventual solution. This method is especially useful for turning non-exact differential equations into exact ones, a key step in finding a general solution.

Boundary Value Problems

Boundary value problems (BVPs) are a class of differential equations where the solution is determined by the values of the function on the borders—or boundaries—of the domain. BVPs are frequently encountered in the physical sciences and engineering when analyzing physical systems that have conditions specified at the boundaries.

The Bessel equation is particularly relevant in BVPs as it appears in scenarios such as heat conduction in cylindrical objects and vibrations of circular membranes, where the boundary conditions can typically be specified on a circle (or circular cylinder). The parameter \( v \) in the Bessel equation is often associated with the modes of vibration in such problems. By solving the Bessel equation with appropriate boundary conditions, one can determine the behavior of the system within the domain. This direct application in physical problems makes the Bessel equation a crucial study in various scientific fields.