Question

Consider the boundary value problem $$ -\left(x y^{\prime}\right)^{\prime}=\lambda x y $$ \(y, y^{\prime}\) bounded as \(x \rightarrow 0, \quad y^{\prime}(1)=0\) (a) Show that \(\lambda_{0}=0\) is an eigenvalue of this problem corresponding to the eigenfunction \(\phi_{0}(x)=1 .\) If \(\lambda>0,\) show formally that the eigenfunctions are given by \(\phi_{n}(x)=\) \(J_{0}(\sqrt{\lambda_{n}} x),\) where \(\sqrt{\lambda_{n}}\) is the \(n\) th positive root (in increasing order) of the equation \(J_{0}^{\prime}(\sqrt{\lambda})=0 .\) It is possible to show that there is an infinite sequence of such roots. (b) Show that if \(m, n=0,1,2, \ldots,\) then $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x=0, \quad m \neq n $$ (c) Find a formal solution to the nonhomogeneous problem $$ \begin{aligned}-\left(x y^{\prime}\right)^{\prime} &=\mu x y+f(x) \\ y, y^{\prime} \text { bounded as } x \rightarrow 0, & y^{\prime}(1)=0 \end{aligned} $$ where \(f\) is a given continuous function on \(0 \leq x \leq 1,\) and \(\mu\) is not an eigenvalue of the corresponding homogeneous problem.

Short answer

Based on the given information, we solved the boundary value problem involving an ordinary differential equation to find the eigenvalues and eigenfunctions. Then, we used them to expand and formally solve the nonhomogeneous problem. We showed that λ₀ = 0 is an eigenvalue corresponding to the eigenfunction φ₀(x) = 1, proved the orthogonality of eigenfunctions, and found the formal solution of the nonhomogeneous problem in terms of the eigenfunctions.

Step-by-step solution

Analyzing the given ODE

Consider the given ODE: $$ -\left(x y^{\prime}\right)^{\prime}=\lambda x y $$ We are given that \(y\) and \(y'\) are bounded as \(x \rightarrow 0\), and \(y^{\prime}(1)=0\). We are asked to show that \(\lambda_{0}=0\) is an eigenvalue corresponding to the eigenfunction \(\phi_{0}(x)=1\).

Calculate the solution for \(\lambda=0\)

Plug \(\lambda=0\) into the ODE, and integrate twice: $$ -\left(x y^{\prime}\right)^{\prime}=0 $$ Integrating once, we get: $$ xy^{\prime}=C_{1} $$ Now integrate for \(y(x)\): $$ y(x)=C_{1}\int\frac{1}{x}dx+C_{2}=C_{1}\ln(x)+C_{2} $$ Now we apply the boundary conditions: \(y\) and \(y^{\prime}\) are bounded as \(x\to 0\). This implies that \(C_{1}=0\), as \(\ln(x)\) would be unbounded as \(x\to0\). Thus, $$ y(x)=C_{2} $$ Now, using the condition \(y^{\prime}(1)=0\), we find that the eigenfunction is \(\phi_{0}(x)=1\) and the corresponding eigenvalue is \(\lambda_{0}=0\).

Formal solution for the eigenfunctions when \(\lambda>0\)

We look for the solutions of the ODE as \(\lambda>0\): $$ -\left(xy^{\prime}\right)^{\prime}=\lambda x y $$ We seek a solution of the form \(y(x)=r(x)J_{0}(kx)\). Substituting this into the ODE, we obtain an equation for the function \(r(x)\): $$ -\left(x \left[r^{\prime}(x)J_{0}(kx)+kr(x)J_{0}^{\prime}(kx)\right]\right)^{\prime}=\lambda x \left[r(x)J_{0}(kx)\right] $$ Now, simplify and divide by \(xJ_{0}(kx)\): $$ -\left[r^{\prime}(x)+kr(x)\frac{J_{0}^{\prime}(kx)}{J_{0}(kx)}\right]^{\prime}=\lambda r(x) $$ This is a first-order ODE for \(r(x)\): $$ r^{\prime}(x)+kr(x)\frac{J_{0}^{\prime}(kx)}{J_{0}(kx)}=\frac{1}{x}r^{\prime\prime}(x) $$ r(x) must be of the form \(A\frac{1}{x}\). Thus, by equating the coefficients of \(r^{\prime\prime}\) and \(r^{\prime}\), we have: $$ k^2=λ $$ So, from the given question, we have: $$\phi_{n}(x)=J_{0}(\sqrt{\lambda_{n}} x)=J_{0}(kx)$$ with the condition that \(J_{0}^{\prime}(k)=0\)

Orthogonality of the eigenfunctions

Now we will prove the orthogonality of eigenfunctions: $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x=0, \quad m \neq n $$ We have \(\phi_{m}(x)=J_{0}(\sqrt{\lambda_{m}} x)\) and \(\phi_{n}(x)=J_{0}(\sqrt{\lambda_{n}} x)\). We can use the orthogonality property of the Bessel functions: $$ \int_0^1 x J_{0}(a x) J_{0}(bx) dx=0, \quad a \neq b $$ Thus, when \(m \neq n\): $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x= 0 $$

Formal solution to the nonhomogeneous problem

Now we try to find the solution of the given nonhomogeneous problem: $$ \begin{aligned}-\left(x y^{\prime}\right)^{\prime} &=\mu x y+f(x) \\\ y, y^{\prime} \text { bounded as } x \rightarrow 0, & y^{\prime}(1)=0 \end{aligned} $$ We need to find the formal solution in terms of the eigenfunctions found in previous steps. We expand the solution \(y(x)\) in terms of the eigenfunctions as: $$ y(x)=\sum_{n=0}^{\infty}c_n \phi_n(x) $$ We can find the coefficients \(c_n\) using the orthogonality property: $$ c_n=\frac{\int_0^1 x \phi_n(x) f(x) dx}{\int_0^1 x \phi_n^2(x) dx} $$ Thus, the formal solution of the nonhomogeneous problem is: $$ y(x)=\sum_{n=0}^{\infty} \frac{\int_0^1 x \phi_n(x) f(x) dx}{\int_0^1 x \phi_n^2(x) dx} \phi_n(x) $$

Key concepts

Understanding Boundary Value Problems

A boundary value problem involves finding a solution to a differential equation that satisfies certain conditions at the boundary of the domain. Unlike initial value problems, where the solution is known at an initial point and extends along an interval, boundary value problems specify conditions at multiple points.
To illustrate, consider the given differential equation \[-(x y')' = \lambda x y \], accompanied by boundary conditions: \( y \) and \( y' \) bounded as \( x \rightarrow 0 \), and \( y'(1) = 0 \). These conditions ensure the behavior of the solution at the endpoints of the interval.

In solving boundary value problems, we often look for eigenvalues and eigenfunctions that satisfy these boundary conditions. This particular dilemma expects determining such values where \( \lambda \) is zero, leading us to the simple constant eigenfunction \( \phi_0(x) = 1 \).
Overall, boundary value problems are fundamental in mathematical modeling, describing phenomena where conditions are known at the limits of the physical system.

Exploring Bessel Functions

Bessel functions appear frequently in solving boundary value problems with circular or cylindrical symmetry. These special functions satisfy Bessel's differential equation, often occurring in problems involving cylindrical coordinates or vibrations of circular membranes.

The given exercise introduces Bessel functions as eigenfunctions, \( \phi_n(x) = J_0(\sqrt{\lambda_n} x) \), where \( J_0 \) is the Bessel function of the first kind of order zero. This reflects the oscillatory nature and zero motion at the edges, seen in their applications, like waves on a drumhead.
In the context of this problem, \( \sqrt{\lambda_n} \) are roots of the equation \( J_0'(\sqrt{\lambda}) = 0 \), providing a sequence of eigenvalues essential for boundary problems.

Bessel functions’ applications extend into physics and engineering, significant in scenarios like heat conduction and electromagnetism.

Orthogonality of Eigenfunctions

Orthogonality of eigenfunctions is a profound concept that simplifies complex boundary value problems. Eigenfunctions associated with distinct eigenvalues are orthogonal over a prescribed interval with a weight function.

For this boundary value problem, orthogonality means that
\[ \int_{0}^{1} x \phi_m(x) \phi_n(x) \, dx = 0, \quad m eq n \],
implies the integral of the product of different eigenfunctions weighted by \( x \) is zero, given they are driven by distinct eigenvalues.
Orthogonal functions enable the expansion of arbitrary functions in terms of these eigenfunctions, much like Fourier series.

This property is crucial for solving nonhomogeneous differential equations, allowing solutions to be expressed as sums over these orthogonal functions, simplifying calculations.

Nonhomogeneous Differential Equations and their Solutions

A nonhomogeneous differential equation includes a term not involving the function or its derivatives, introducing a 'forcing' aspect through an arbitrary function, \( f(x) \). The task is to discover a solution satisfying the original equation plus this additional driving force.

The general form includes terms like:
\[ -(x y')' = \mu x y + f(x) \], with given boundary conditions.
In this context, a formal solution involves expressing \( y(x) \) as a series of eigenfunctions, \( \phi_n(x) \), adding coefficients altering their individual amplitudes based on the function \( f(x) \).

• We use orthogonality, calculating coefficients \( c_n \) via
  \[ c_n = \frac{\int_0^1 x \phi_n(x) f(x) \ dx}{\int_0^1 x \phi_n^2(x) \ dx} \].


• This transforms our specific problem into a superposition of those eigenfunctions, effectively tackling the nonhomogeneous aspect.

Solving such equations extends into fields where systems are influenced by external forces, importantly in physics and engineering.