Question

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

Short answer

Answer: The Green's function for the given boundary-value problem is: $$ G(x, \xi) = \begin{cases} \ln{x}, & \text{if}\ x \leq \xi, \\ 0, & \text{if}\ x > \xi. \end{cases} $$

Step-by-step solution

Identify the Homogeneous ODE

First, let's write down the homogeneous version of the given ODE, which can be obtained by replacing the forcing term \(f(x)\) with zero: $$ xy'' + y' = 0 $$

Solve the Homogeneous ODE

We solve the homogeneous ODE using the method of separation of variables. Dividing the equation by \(y'\) and integrating both sides, we get: $$ \int\frac{dy'}{y'} = -\int\frac{dx}{x} $$ Upon integrating, we get: $$ \ln|y'| = -\ln|x| + \ln{c_1} $$ which simplifies to: $$ y' = \frac{c_1}{x} $$ Now, we integrate again to get \(y(x)\): $$ y(x) = c_1 \ln{x} + c_2 $$ Since we have a second-order ODE, we expect to have two linearly independent solutions. Let the solutions be \(y_1(x) = \ln{x}\) and \(y_2(x) = 1\) with the corresponding constants \(c_1 = 1\) and \(c_2 = 0\), respectively.

Formulate Green's Function

Recall that the Green's function is expressed as: $$ G(x, \xi) = \begin{cases} A(\xi)y_1(x), & \text{if}\ x \leq \xi, \\ B(\xi)y_2(x), & \text{if}\ x > \xi. \end{cases} $$ where \(A(\xi)\) and \(B(\xi)\) are functions to be determined based on the boundary conditions. Using the boundary condition \(y(1) = 0\), we have: $$ 0 = A(\xi) y_1(1) + B(\xi) y_2(1) = A(\xi) \ln 1 + B(\xi) \implies B(\xi) = 0 $$ So our Green's function becomes: $$ G(x, \xi) = \begin{cases} A(\xi)y_1(x), & \text{if}\ x \leq \xi, \\ 0, & \text{if}\ x > \xi. \end{cases} $$ Now, to determine \(A(\xi)\), we have the jumps in the derivative of the Green's function due to the mixed boundary condition: $$y'(0^+) - y'(0^-) = G_x(0^+, \xi) - G_x(0^-, \xi) = \lim_{x\rightarrow 0}(A(\xi)y_1'(x)) - 0 = -A(\xi) = -1$$ Therefore, \(A(\xi) = 1\). So, the Green's function is now defined as: $$ G(x, \xi) = \begin{cases} \ln{x}, & \text{if}\ x \leq \xi, \\ 0, & \text{if}\ x > \xi. \end{cases} $$

Verify Green's Function

Finally, let's verify if our Green's function satisfies the given boundary-value problem. We plug in the Green's function into the ODE and check if it produces the required conditions: 1. The ODE is satisfied in the \((0,1)\) interval. 2. The boundary conditions at \(x=1\) and as \(x\rightarrow 0\) are satisfied. For \(0 < x \leq \xi\), we have \(G(x, \xi) = \ln{x}\), so \(G'(x, \xi) = \frac{1}{x}\), and \(G''(x, \xi) = -\frac{1}{x^2}\). Substituting these into the ODE: $$ x\left(-\frac{1}{x^2}\right) + \frac{1}{x} = 0 $$ which is true, satisfying the ODE in the required range. For the boundary conditions, at \(x = 1\), we have \(G(1, \xi) = \ln{1} = 0\), which satisfies the first boundary condition. As \(x\rightarrow 0\), \(|\ln x|\rightarrow\infty\), but \(\ln x\) is multiplied by \(A(\xi) = 1\). Therefore, the second boundary condition is also satisfied. Thus, the Green's function found is correct for the given problem.

Key concepts

Boundary-value problems

A boundary-value problem is a type of differential equation that specifies the behavior of a function along the boundaries of its domain. Unlike initial value problems, which give conditions at only one point, boundary-value problems provide conditions at two or more points. In physics and engineering, these problems are crucial because they often model real-world phenomena, such as the shape of a bridge that must remain within certain limits both at the ends and along its length.

• Boundary conditions are essential as they help narrow down the infinite possible solutions of a differential equation to the most applicable one.
• The conditions mentioned in the exercise are that the solution must fulfill two boundary conditions: it must be finite as \(x \rightarrow 0\) and vanish at \(x = 1\).
• In our specific problem, finding the Green's function helps determine how the boundary conditions influence solutions of the corresponding inhomogeneous differential equation.

Understanding boundary values not only simplifies complex real-world applications but also makes the theoretical aspects of differential equations easier to handle.

Homogeneous ordinary differential equations

Homogeneous ordinary differential equations are those in which all terms are a function of the unknown variable and its derivatives. Importantly, there is no stand-alone function or constant term. This makes them particularly useful for finding solutions that naturally satisfy linear conditions.

• The given homogeneous equation is \( xy'' + y' = 0 \). It is homogeneous because there is no independent function of \(x\) on the right-hand side.
• Solving homogeneous differential equations usually results in a set of solutions that can be superposed to form a general solution for the problem.
• This is important because it provides a foundation upon which solutions to non-homogeneous equations, or those with additional forcing terms, can be built.

Understanding homogeneous solutions is a stepping stone toward tackling more complex scenarios, such as when external forces or inputs are added, which often leads us to non-homogeneous cases.

Method of separation of variables

The method of separation of variables is a powerful technique used to solve differential equations by separating the variables into distinct functions of each independent variable involved. This makes differential equations more manageable and often helps convert them into simpler forms we can directly integrate.

• In our exercise, separation of variables allows us to handle the equation \( xy'' + y' = 0 \) by rearranging it and integrating it in terms of \(y'\) versus \(x\).
• The integration of \( \int \frac{dy'}{y'} = -\int \frac{dx}{x} \) leads us through a process that simplifies the problem significantly.
• After separation and integration, we identify and express the solution as a product or sum of functions only involving a single variable.

This method is particularly useful for linear equations and serves as a classic approach to solving many types of differential equations efficiently.

Linear independence of solutions

Linear independence is a concept that indicates whether a set of solutions to a differential equation can be expressed as linear combinations of one another. If solutions are linearly independent, they provide unique parts of the general solution, essential for constructing varied solutions.

• For the given second-order differential equation, we expect two linearly independent solutions: \( y_1(x) = \ln{x} \) and \( y_2(x) = 1 \).
• Linear independence means that these solutions cannot be written as scalar multiples of each other, which is checked using the Wronskian determinant. A non-zero Wronskian confirms independence.
• Having two linearly independent solutions is critical for constructing the general solution. It ensures that whatever boundary conditions are applied, we can adjust constants to fit precisely.

Understanding linear independence not only helps differentiate between solutions but also validates that we have enough independent solutions to fully describe the system under study.