Question

Find the generalized Green's function for the boundary-value problem $$ \begin{aligned} x y^{\prime \prime}+y^{\prime} &=-f(x) \\ y(1)=0, & \lim _{x \rightarrow 0}|y(x)|<\infty \end{aligned} $$

Short answer

Question: Find the generalized Green's function for the following boundary-value problem involving a second-order linear ordinary differential equation (ODE) with variable coefficients: $$ x y'' + y' = -f(x) $$ Subject to the boundary conditions: \(y(1) = 0\) and \(\lim_{x \to 0} |y(x)| < \infty\). Answer: The generalized Green's function for the given boundary-value problem is given by: $$ y(x) = \int_0^1 \xi f(\xi) d\xi - x \int_0^1 f(\xi) d\xi $$

Step-by-step solution

Write down the homogeneous ODE corresponding to the given equation

A homogeneous ODE corresponding to the given equation can be obtained by removing the forcing function (i.e., replacing \(-f(x)\) with \(0\)): $$ x y'' + y' = 0 $$

Solve the homogeneous ODE using an integrating factor

Let's multiply the homogeneous ODE by an integrating factor, in this case, \(\frac{1}{x}\). Then we have: $$ y'' + \frac{1}{x} y' = 0 $$ Now we observe that the left side of the equation is an exact derivative of a function of x: $$ \frac{d}{dx}\left(y' + \frac{y}{x}\right) = 0 $$ Integrating both sides, we have: $$ y' + \frac{y}{x} = C $$ where \(C\) is a constant of integration.

Obtain a particular solution satisfying the boundary condition at \(x = 1\)

Applying the boundary condition, \(y(1) = 0\), we can find the value of the constant \(C\). We have: $$ y'(1) + \frac{y(1)}{1} = C ⟹ C = 0 $$ The homogeneous ODE becomes: $$ y' + \frac{1}{x} y = 0 $$ This is a first-order linear ODE and can be solved using another integrating factor. Multiplying both sides of the equation by \(x\), we get: $$ xy' + y = 0 $$ Now observe that the left side of the equation is an exact derivative: $$ \frac{d}{dx}\left(xy\right) = 0 $$ Integrating both sides, we have: $$ xy = D $$ where \(D\) is a constant of integration. So, we have found a particular solution of the problem: $$ y = \frac{D}{x} $$

Find the natural Green's function satisfying the boundary condition at \(x = 0\)

Now let's consider the natural Green's function \(G(x, \xi)\), which corresponds to the Dirac delta forcing function, i.e. \(-f(x) = \delta(x - \xi)\). The Green's function satisfies the same boundary-value problem as before: $$ x G'' + G' = \delta(x - \xi) $$ with \(G(1,\xi) = 0\) and \(\lim_{x \to 0} |G(x,\xi)| < \infty\). Since we already have found a particular solution, \(y = \frac{D}{x}\), which satisfies the condition at \(x=1\), let's express \(G(x,\xi)\) in the following form: $$ G(x,\xi) = \frac{A(\xi)}{x} $$ Substituting this expression into the equation for the Green's function, we obtain: $$ -x\frac{d^2}{dx^2}\left(\frac{A(\xi)}{x}\right) + \frac{d}{dx}\left(\frac{A(\xi)}{x}\right) = \delta(x - \xi) $$ Integration by parts and use of the Dirac delta properties yield: $$ A(\xi) = -x^2 $$ Thus, the natural Green's function is: $$ G(x,\xi) = \frac{-x^2}{x} = -x $$

Find the generalized Green's function

In order to find the generalized Green's function (i.e., taking into account the forcing function \(-f(x)\)), we will reuse the natural Green's function: $$ y(x) = -\int_0^1 G(x,\xi) f(\xi) d\xi $$ Substituting the found Green's function, and applying the boundary condition \(y(1) = 0\), we have: $$ y(x) = \int_0^1 \xi f(\xi) d\xi - x \int_0^1 f(\xi) d\xi $$ So, we have found the generalized Green's function for the given boundary-value problem.

Key concepts

Boundary-value problem

A boundary-value problem (BVP) is a type of differential equation along with a set of additional constraints, known as boundary conditions. These conditions are specified at the boundaries of the interval where the solution is defined. In our problem, the boundary conditions are given as \( y(1) = 0 \) and the condition that the solution remains finite as \( x \) approaches 0. These conditions specify the behavior of the solution at particular points and are crucial for finding a unique solution.

When solving a BVP, we aim to find a function that satisfies the differential equation and these boundary conditions. Green's functions play a vital role in solving BVPs as they offer a method to determine solutions for linear differential equations subject to boundary conditions.

Homogeneous ordinary differential equation

A homogeneous ordinary differential equation (ODE) is an equation involving derivatives of a function but without any additional terms (called forcing functions). The homogeneous version of a given ODE is formed by setting the forcing function to zero.

For example, the homogeneous ODE from our original problem is \( x y'' + y' = 0 \). Solving this equation involves finding solutions where these derivatives satisfy the equation exactly without any additional external input (like \(-f(x)\)).

These solutions are essential since they form the basis for constructing solutions to the non-homogeneous problem, which includes additional terms or forcing functions.

Integrating factor

An integrating factor is a function used to simplify the process of solving a differential equation. It is particularly helpful for solving linear first-order differential equations. The goal of using an integrating factor is to transform a non-exact equation into an exact one, which can then be easily integrated.

In our exercise, the use of integrating factor \( \frac{1}{x} \) helped transform the homogeneous ODE into a form that is easier to solve: \( y'' + \frac{1}{x} y' = 0 \). By doing this, it becomes apparent that the equation is actually the derivative of something simpler, which allows it to be integrated straightforwardly. This method is a powerful technique commonly used in differential equations to tackle otherwise complex problems.

Dirac delta function

The Dirac delta function is a mathematical construct used to model an idealized point source or impulse. It's a unique function with an infinitely high peak at a single point, and an area under the curve equal to one. In the context of differential equations, the delta function is used to represent concentrated forces or sources.

In our Green's function problem, the Dirac delta function appears as \( \delta(x - \xi) \), representing a point force at position \( \xi \). The presence of this function in an equation indicates that we're dealing with a singular disturbance at a precise location.

This property is vital in creating Green's functions, which are used to describe how the system responds to such singular inputs, thereby providing comprehensive solutions to boundary-value problems.