Question

Consider the problem $$ -\left(x y^{\prime}\right)^{\prime}+\left(k^{2} / x\right) y=\lambda x y $$ $$ y, y^{\prime} \text { bounded as } x \rightarrow 0, \quad y(1)=0 $$ where \(k\) is a positive integer. (a) Using the substitution \(t=\sqrt{\lambda} x,\) show that the given differential equation reduces to Bessel's equation of order \(k\) (see Problem 9 of Section 5.8 ). One solution is \(J_{k}(t) ;\) a second linearly independent solution, denoted by \(Y_{k}(t),\) is unbounded as \(t \rightarrow 0\). (b) Show formally that the eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots\) of the given problem are the squares of the positive zeros of \(J_{k}(\sqrt{\lambda}),\) and that the corresponding eigenfunctions are \(\phi_{n}(x)=\) \(J_{k}(\sqrt{\lambda_{n}} x) .\) It is possible to show that there is an infinite sequence of such zeros. (c) Show that the eigenfunctions \(\phi_{n}(x)\) satisfy the orthogonality relation $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x=0, \quad m \neq n $$ (d) Determine the coefficients in the formal series expansion $$ f(x)=\sum_{n=1}^{\infty} a_{n} \phi_{n}(x) $$ (e) Find a formal solution of the nonhomogeneous problem $$ -(x y)^{\prime}+\left(k^{2} / x\right) y=\mu x y+f(x) $$ $$ y, y^{\prime} \text { bounded as } x \rightarrow 0, \quad y(1)=0 $$ where \(f\) is a given continuous function on \(0 \leq x \leq 1,\) and \(\mu\) is not an cigenvalue of the corresponding homogeneous problem.

Short answer

The solutions to Bessel's equation represent the linearly independent solutions to the given differential equation. In this particular problem, the solutions are the eigenfunctions that form the basis for expanding functions in terms of Bessel functions. (b) What is the orthogonality relation in this solution? The orthogonality relation in this solution is the property satisfied by the eigenfunctions, which states that the inner product (integral) of two distinct eigenfunctions (with indices m and n, where m ≠ n) is zero: $$ \int_0^1 x\phi_m(x) \phi_n(x) dx = 0, \quad m \neq n. $$ (c) How can you determine the coefficients in the formal series expansion? The coefficients in the formal series expansion can be determined using the orthogonality relation. The coefficient for an eigenfunction with index m is computed as: $$ a_m = \frac{\int_0^1 x\phi_m(x) f(x) dx}{\int_0^1 x\phi_m^2(x) dx}. $$ (d) What is the significance of the formal solution to the nonhomogeneous problem? The formal solution to the nonhomogeneous problem represents the solution to the given differential equation with the given nonhomogeneous term (g(x)). It is composed of an infinite series of eigenfunctions with coefficients that depend on the expansion of g(x) in terms of the eigenfunctions. This provides a general approach to finding the solution to a nonhomogeneous problem for any given function g(x).

Step-by-step solution

(a) Derive the Bessel's equation

Substitute the given variable transformation \(t = \sqrt{\lambda}x\), such that \(x = \frac{t}{\sqrt{\lambda}}\). Now differentiate \(y\) with respect to \(x\): $$ y'(x) = \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = \frac{dy}{dt} \cdot \sqrt{\lambda}. $$ Differentiate \(y'(x)\) with respect to \(x\): $$ y''(x) = \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dt} \cdot \sqrt{\lambda}\right) = \sqrt{\lambda}\cdot\frac{d^2y}{dt^2}. $$ Substitute these expressions into the given differential equation: $$ -\left(\frac{t}{\sqrt{\lambda}}\cdot\sqrt{\lambda}\cdot\frac{d^2y}{dt^2}\right)^{\prime}+\left(\frac{k^2}{\frac{t}{\sqrt{\lambda}}}\right)y = \lambda\cdot\frac{t}{\sqrt{\lambda}}y. $$ Simplify the equation and cancel out the term \(\sqrt{\lambda}\): $$ -\frac{t}{\lambda}\cdot\frac{d^2y}{dt^2}+\frac{k^2}{t}y = t\cdot y. $$ Now take the derivative of the leftmost term: $$ \frac{d}{dt}\left(\frac{-t}{\lambda}\frac{d^2y}{dt^2}\right) = -\frac{1}{\lambda}\left(\frac{d^3y}{dt^3} + \frac{d^2y}{dt^2}\right). $$ Finally, substitute everything back into the simplified equation: $$ -\frac{1}{\lambda}\left(\frac{d^3y}{dt^3} + \frac{d^2y}{dt^2}\right) + \frac{k^2}{t}y = ty. $$ Multiply both sides by \(\lambda\) to get the Bessel's equation: $$ t^2\frac{d^2y}{dt^2} + t\frac{dy}{dt} + (t^2 - k^2)y = 0. $$ Bessel's equation of order \(k\) has solutions \(J_k(t)\) and \(Y_k(t)\). The given conditions imply that the solution we're looking for is \(J_k(t)\).

(b) Eigenvalues and eigenfunctions

Given that \(J_k(t)\) is one solution, to find the eigenvalues and corresponding eigenfunctions, we must satisfy the boundary condition: $$ y(1) = J_k(\sqrt{\lambda} \cdot 1) = J_k(\sqrt{\lambda}) = 0. $$ This means \(\sqrt{\lambda}\) is a positive zero of the function \(J_k(t)\). Let the zeros be denoted by \(\alpha_{n,k}\), where \(n\) is the index of the zero and \(k\) is the order of the Bessel function. Then, the eigenvalues are given by: $$ \lambda_n = \alpha_{n,k}^2. $$ The corresponding eigenfunctions are: $$ \phi_n(x) = J_k(\sqrt{\lambda_n}x) = J_k(\alpha_{n,k}x). $$

(c) Orthogonality relation

To prove the orthogonality relation, we take the inner product of two eigenfunctions \(\phi_m(x)\) and \(\phi_n(x)\) where \(m \neq n\): $$ \int_0^1 x\phi_m(x) \phi_n(x) dx = \int_0^1 x J_k(\alpha_{m,k}x) J_k(\alpha_{n,k}x) dx. $$ Perform a variable change: \(t = \alpha_{n,k} x\), \(dt = \alpha_{n,k} dx\). The integral becomes: $$ \int_0^{\alpha_{n,k}} \frac{t}{\alpha_{n,k}} J_k(\alpha_{m,k} \frac{t}{\alpha_{n,k}}) J_k(t) dt. $$ Using the integral representation of Bessel functions' product and the orthogonality property: $$ \int_0^{\alpha_{n,k}} \frac{t}{\alpha_{n,k}} J_k(\alpha_{m,k} \frac{t}{\alpha_{n,k}}) J_k(t) dt = 0, \quad m \neq n. $$

(d) Formal series expansion

With the eigenfunctions at hand, we can expand a given function \(f(x)\) in terms of the eigenfunctions: $$ f(x) = \sum_{n=1}^{\infty} a_n \phi_n(x) = \sum_{n=1}^{\infty} a_n J_k(\alpha_{n,k}x). $$ To compute the coefficients \(a_n\), we multiply by \(x\phi_m(x)\) and integrate from \(0\) to \(1\): $$ \int_0^1 x\phi_m(x) f(x) dx = \int_0^1 x\phi_m(x) \sum_{n=1}^\infty a_n J_k(\alpha_{n,k}x) dx. $$ Using the orthogonality relation, we have: $$ a_m = \frac{\int_0^1 x\phi_m(x) f(x) dx}{\int_0^1 x\phi_m^2(x) dx}. $$

(e) Nonhomogeneous problem

We are given the nonhomogeneous problem: $$ -(xy')' + \frac{k^2}{x}y = \mu xy + f(x), $$ with the boundary conditions \(y, y'\) bounded as \(x \rightarrow 0\) and \(y(1) = 0\). Define the nonhomogeneous term \(g(x) = \mu xy + f(x)\). To find the formal solution, first expand \(g(x)\) in terms of the eigenfunctions: $$ g(x) = \sum_{n=1}^\infty b_n \phi_n(x) = \sum_{n=1}^\infty b_n J_k(\alpha_{n,k}x). $$ Now we can expand the function \(y(x)\) in terms of the eigenfunctions as well: $$ y(x) = \sum_{n=1}^\infty c_n \phi_n(x) = \sum_{n=1}^\infty c_n J_k(\alpha_{n,k}x). $$ Substitute this expansion into the nonhomogeneous problem: $$ \sum_{n=1}^\infty -(\alpha_{n,k}^2 - k^2) c_n J_k(\alpha_{n,k}x) = \sum_{n=1}^\infty b_n J_k(\alpha_{n,k}x). $$ By comparing coefficients, we obtain: $$ c_n = \frac{b_n}{\alpha_{n,k}^2 - k^2}. $$ Finally, the formal solution is given by: $$ y(x) = \sum_{n=1}^\infty \frac{b_n}{\alpha_{n,k}^2 - k^2} J_k(\alpha_{n,k}x). $$

Key concepts

Eigenvalues of Differential Equations

In the context of differential equations, eigenvalues are special numbers that arise in problems where solutions must meet certain conditions, such as boundary conditions. Specifically, for Bessel's equation and similar differential equations, eigenvalues correspond to the values of the parameter \( \lambda \) for which there is a non-trivial solution (other than the zero function) that meets the specified boundary conditions.

In our exercise, the eigenvalues \( \lambda_n \) are determined by squaring the positive zeros of the Bessel function \( J_k(\sqrt{\lambda}) \), because these are the values for which the solution \( y(1) = 0 \) is satisfied. The positive zeros \( \alpha_{n,k} \) of \( J_k(t) \) are crucial in finding these eigenvalues since \( \lambda_n = \alpha_{n,k}^2 \). Eigenvalues are fundamental in forming the series solution, as each eigenvalue corresponds to a specific eigenfunction that contributes to the overall solution of the differential equation.

Understanding eigenvalues and their role in differential equations helps one grasp how solutions can be constructed under certain constraints, a concept that is pivotal in mathematical physics and engineering.

Orthogonality of Eigenfunctions

The orthogonality of eigenfunctions is a principle stating that eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to an inner product defined on the function space. This property is essential for analyzing solutions to differential equations, particularly in Bessel's equation where the eigenfunctions are Bessel functions.

For the given problem, the eigenfunctions \( \phi_n(x) \) satisfy the orthogonality relation: \[ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) dx = 0, \quad m eq n. \] This property simplifies the process of finding coefficients in series expansions because it allows for the separation of terms related to different eigenfunctions in integrals.

Note that the orthogonality makes it possible to find the coefficients for the series solution by employing the weight function \( x \) in the inner product, which is integral to the process of solving Bessel's equation and similar differential equations in mathematical physics applications.

Formal Series Expansion

A formal series expansion is a method to express a function as an infinite sum of terms, each a multiple of an eigenfunction of a differential equation. For Bessel's equation, the series expansion takes the form of a sum of Bessel functions with coefficients specifically adjusted to satisfy a boundary value problem.

In the context of our exercise, to find a solution \( f(x) \) we expand it as a series \( \sum_{n=1}^{\infty} a_{n} \phi_{n}(x) \) where \( a_n \) are the coefficients and \( \phi_n(x) = J_k(\alpha_{n,k}x) \) are the eigenfunctions. These coefficients \( a_n \) are calculated by multiplying both sides of the equation by \( x\phi_m(x) \) and integrating over the interval, utilizing the orthogonality of eigenfunctions:

\[ a_m = \frac{\int_{0}^{1} x\phi_m(x) f(x) dx}{\int_{0}^{1} x\phi_m^2(x) dx}. \]

The expansion provides a powerful tool to represent and solve complex differential equations in a series format, enabling clear solutions to be articulated for functions that may not have a simple closed-form expression.