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Chapter 9: Problem 10

Bessel's equation \(y^{\prime \prime}+\frac{1}{t} y^{\prime}+\left(1-\frac{n^{2}}{t^{2}}\right) y=0\)

Short Answer

Expert verified
The solution to the given Bessel's differential equation of order \(n\) is \(y(t)=C_1J_n(t) + C_2Y_n(t)\)

Step by step solution

01

Problem Analysis

The given differential equation is Bessel's equation of order \(n\): \[y''+\frac{1}{t}y'+\left(1-\frac{n^{2}}{t^{2}}\right)y=0\] The solutions of Bessel's differential equation in general form are given by: \[y(t)=C_1J_n(t) + C_2Y_n(t)\] where\(J_n(t)\) and \(Y_n(t)\) are Bessel functions of the first and second kind respectively and order \(n\), and \(C_1\) and \(C_2\) are arbitrary constants.
02

Writing the Bessel鈥檚 Equation Solution

Bessel's equation solutions are expressible in terms of the Bessel Function of first kind and the second kind. This formulation requires the knowledge that the solution of Bessel鈥檚 equation is given by Bessel鈥檚 functions. \[y(t)=C_1J_n(t) + C_2Y_n(t)\] In the solution, \(C_1\) and \(C_2\) are constants, \(J_n\) and \(Y_n\) respectively are the Bessel function of the first kind and the second kind of order n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations form the cornerstone of dynamic systems and their analyses. They are mathematical equations that relate some function with its derivatives. In the context of Bessel's equation, it represents a second-order linear differential equation. In the broader scope of mathematical analysis and applications, differential equations allow us to model growth decays, oscillations, and even complex phenomena like heat conduction and wave propagation. They not only describe how a system evolves over time but also enable us to predict future behavior given certain conditions.

Understanding differential equations involves recognizing the order, linearity, and the function being derived. In our case, Bessel鈥檚 equation is a second-order equation because it contains the second derivative of the function. It is also a homogeneous linear differential equation, meaning that it can be set to zero, and each of its terms is proportional to the function or its derivatives.
Bessel Functions
Bessel functions are solutions to Bessel's differential equation that appear in a wide array of physical problems, such as heat conduction, wave propagation in cylindrical or spherical coordinates, and vibrations of a circular membrane. Specifically, they are defined as the canonical solutions \(J_n(x)\) of the first kind and \(Y_n(x)\) of the second kind, where \(n\) represents the order of the function.

These two types of Bessel functions exhibit oscillatory behavior and are particularly useful in problems with circular symmetry. For instance, \(J_n(x)\) is finite at the origin while \(Y_n(x)\) is singular at the origin. A key aspect of Bessel functions is their orthogonality, which allows them to form a complete set over their defined domain鈥攁 property incredibly useful in solving boundary value problems.
Mathematical Analysis
Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are used to model and solve problems that involve precise levels of change and quantities.

Within mathematical analysis, Bessel's equation and the related functions fall under the topic of differential equations and special functions. By studying these specific aspects of analysis, advanced mathematical tools are developed to tackle physical problems that are modelled by differential equations. Analysis ensures that the solution of Bessel鈥檚 equation, involving Bessel functions, is well-defined and behaves according to the expected properties in a given context.
Ordinary Differential Equations
Ordinary differential equations (ODEs), like Bessel's equation, involve functions of only one independent variable and their derivatives. These equations are classified according to their order, linearity, and whether they are homogeneous or non-homogeneous. A key objective in solving an ODE is to find a function that satisfies the equation, often requiring a combination of analytical techniques and computational methods.

For the Bessel's equation, the standard approach involves recognizing it as a Sturm-Liouville problem, which facilitates the finding of solutions that are Bessel functions. This strategy is a fundamental one within the realm of ODEs, and mastering it is crucial for students aiming to apply mathematical methods to physical and engineering problems.

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Most popular questions from this chapter

17\. $$e^{2 t} \left[ \begin{array}{l}{1} \\ {0} \\ {5}\end{array}\right], \quad e^{2 t} \left[ \begin{array}{r}{1} \\ {1} \\ {-1}\end{array}\right], \quad e^{3 t} \left[ \begin{array}{l}{0} \\ {1} \\ {0}\end{array}\right]$$

26\. Verify that the vector functions $$\mathbf{x}_{1}=\left[ \begin{array}{c}{e^{3 t}} \\ {0} \\ {e^{3 t}}\end{array}\right], \quad \mathbf{x}_{2}=\left[ \begin{array}{r}{-e^{3 t}} \\\ {e^{3 t}} \\ {0}\end{array}\right], \quad \mathbf{x}_{3}=\left[ \begin{array}{c}{-e^{-3 t}} \\ {-e^{-3 t}} \\ {e^{-3 t}}\end{array}\right]$$ are solutions to the homogeneous system $$\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}=\left[ \begin{array}{rrr}{1} & {-2} & {2} \\ {-2} & {1} & {2} \\ {2} & {2} & {1}\end{array}\right] \mathbf{x}$$ $$(-\infty, \infty),$$ $$\mathbf{x}_{p}=\left[ \begin{array}{c}{5 t+1} \\ {2 t} \\ {4 t+2}\end{array}\right]$$ is a particular solution to \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}+\mathbf{f}(t),\) where \(\mathbf{f}(t)=\operatorname{col}(-9 t, 0,-18 t) .\) Find a general solution to \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}+\mathbf{f}(t).\)

(a) Show that the Cauchy-Euler equation \(a t^{2} y^{\prime \prime}+b t y^{\prime}+c y=0 \quad\) can \(\quad\) be written as a Cauchy-Euler system $$ t \mathbf{x}^{\prime}=\mathbf{A x} $$ with a constant coefficient matrix \(\mathbf{A},\) by setting \(x_{1}=y / t\) and \(x_{2}=y^{\prime}\) (b) Show that for \(t>0\) any system of the form \((25)\) with A an \(n \times n\) constant matrix has nontrivial solutions of the form \(\mathbf{x}(t)=t^{r} \mathbf{u}\) if and only if \(r\) is an eigenvalue of \(\mathbf{A}\) and \(\mathbf{u}\) is a corresponding eigenvector.

31\. Show that $$\left| \begin{array}{ll}{t^{2}} & {t|t|} \\ {2 t} & {2|t|}\end{array}\right|=0$$ on \((-\infty, \infty),\) but that the two vector functions $$\left[ \begin{array}{l}{t^{2}} \\ {2 t}\end{array}\right], \quad \left[ \begin{array}{l}{t|t|} \\ {2|t|}\end{array}\right]$$ are linearly independent on \((-\infty, \infty).\)

To find a general solution to the system $$\mathbf{x}^{\prime}(t)=\left[ \begin{array}{rr}{0} & {1} \\ {-2} & {3}\end{array}\right] \mathbf{x}(t)+\mathbf{f}(t), \quad\( where \)\quad \mathbf{f}(t)=\left[ \begin{array}{l}{e^{t}} \\ {0}\end{array}\right]$$ proceed as follows: (a) Find a fundamental solution set for the corresponding homogeneous system. (b) The obvious choice for a particular solution would be a vector function of the form \(\mathbf{x}_{p}(t)=e^{t} \mathbf{a} ;\) how ever, the homogeneous system has a solution of this form. The next choice would be \(\mathbf{x}_{p}(t)=t e^{\prime} \mathbf{a}\) . Show that this choice does not work. (c) For systems, multiplying by \(t\) is not always sufficient. The proper guess is $$\mathbf{x}_{p}(t)=t e^{l} \mathbf{a}+e^{\prime} \mathbf{b}$$ Use this guess to find a particular solution of the given system. (d) Use the results of parts (a) and (c) to find a general solution of the given system.
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