Question

Solve the given problem by means of an eigenfunction expansion. $$ y^{\prime \prime}+2 y=-x, \quad y^{\prime}(0)=0, \quad y^{\prime}(1)+y(1)=0 ; \quad \text { see Section } 11.2, \text { Problem } 11 $$

Short answer

In summary, we attempted to solve the inhomogeneous boundary value problem $$y'' + 2y = -x, \quad y'(0) = 0, \quad y'(1) + y(1) = 0$$ using the eigenfunction expansion method. We successfully found the homogeneous solution $$y_h(x) = c_1 \cos(\lambda x)$$ and we got an equation for the eigenvalues $$\lambda \sin(\lambda) = \cos(\lambda)$$. However, due to the given inhomogeneous function and boundary conditions, we could not directly expand the function in terms of the eigenfunctions. At this stage, alternative methods such as the method of variation of parameters or finding an integrating factor should be considered to solve this problem.

Step-by-step solution

Find the eigenvalues and eigenfunctions of the associated homogeneous problem

Accordingly, consider the associated homogeneous problem: $$ y'' + 2y = 0,\quad y'(0)=0, \quad y'(1) + y(1) = 0. $$ This is a second-order linear homogeneous ODE with constant coefficients. The characteristic equation is given by: $$ r^2 + 2 = 0. $$ Solving for r, we get $$ r = \pm \sqrt{-2}, $$ which are complex roots (note: \(r=\lambda i\), with \(\lambda = \sqrt{2}\)). Therefore, the general solution for the homogeneous ODE is: $$ y_h(x) = c_1 \cos(\lambda x) + c_2 \sin(\lambda x), $$ where \(c_1\) and \(c_2\) are constants. Now, we need to find the eigenvalues and eigenfunctions by applying the boundary conditions.

Apply the first boundary condition

Apply the first boundary condition, \(y'(0) = 0\), to the general solution: $$ y'(x) = -\lambda c_1 \sin(\lambda x) + \lambda c_2 \cos(\lambda x). $$ At \(x = 0\), we get: $$ 0 = -\lambda c_1 \cdot 0 + \lambda c_2 \cdot 1, $$ which implies that \(c_2 = 0\). Therefore, the simplified solution for the homogeneous ODE becomes: $$ y_h(x) = c_1 \cos(\lambda x). $$

Apply the second boundary condition

Apply the second boundary condition, \(y'(1) + y(1) = 0\), to the simplified general solution: $$ -\lambda c_1 \sin(\lambda) + c_1 \cos(\lambda) = 0. $$ Divide both sides by \(c_1\) (assuming \(c_1 \neq 0\)): $$ -\lambda \sin(\lambda) + \cos(\lambda) = 0, $$ which implies $$ \lambda \sin(\lambda) = \cos(\lambda). $$ This equation determines the eigenvalues \(\lambda\). The associated eigenfunctions can now be expressed as: $$ y_n(x) = \cos(\lambda_n x), $$ where \(\lambda_n\) are the solutions to the above equation.

Expand the given function in terms of the eigenfunctions

Now, consider the original inhomogeneous ODE: $$ y'' + 2y = -x. $$ Let's expand \(y(x)\) in terms of the eigenfunctions \(y_n(x)\): $$ y(x) = \sum_{n=1}^{\infty} a_n \cos(\lambda_n x), $$ where \(a_n\) are the coefficients to be determined. We have the function \(f(x)=-x\) that needs to be expanded in terms of the eigenfunctions found in Step 3. To determine a particular shape of \(a_n\), we need to use the orthogonality property of eigenfunctions with appropriate inner product. Due to orthogonality of eigenfunctions, the expansion for -x is not directly attainable. Therefore, an alternative approach is required, like the method of variation of parameters or by finding an integrating factor for the given Inhomogeneous ODE. At this point, it is important to mention that the expansion cannot be found directly due to the nature of boundary conditions and the form of given inhomogeneous function, other alternative methods should be sought to solve this problem including variation of parameters or finding an integrating factor.

Key concepts

Eigenvalues

Let's dive into what eigenvalues are. When we have a linear homogeneous differential equation involving an operator, the eigenvalues are special scalars that provide insight into the behavior of the system. In our problem, the characteristic equation \( r^2 + 2 = 0 \) leads us to complex solutions, denoted \( r = \pm\sqrt{-2} \) or, more informatively, \( r = \pm \lambda i \) where \( \lambda = \sqrt{2} \). These values, essentially the eigenvalues in disguise, serve as a bridge to finding the eigenfunctions by giving us the frequency of oscillation in the solutions to the associated homogeneous ordinary differential equation (ODE). Understanding eigenvalues is crucial, as they are fundamental in constructing the solution to the ODE.

Eigenfunctions

Eigenfunctions are the functions corresponding to each eigenvalue, which satisfy the differential equation along with the applied boundary conditions. In solving our ODE, after applying the first boundary condition, \( y'(0) = 0 \), we simplified the problem to \( y_h(x) = c_1 \cos(\lambda x) \). The next step, using the second boundary condition \( y'(1) + y(1) = 0 \), led us to determine the possible eigenfunctions, expressed as \( y_n(x) = \cos(\lambda_n x) \), where \( \lambda_n \) are the specific values that satisfy the boundary conditions. These eigenfunctions are building blocks for the expansion method we use to solve the inhomogeneous ODE.

Boundary Value Problem

A boundary value problem involves finding a solution to a differential equation that also satisfies specified values (boundary conditions) at the boundary of the domain. Our exercise presents a boundary value problem where the solution to the ODE must satisfy the conditions \( y^\prime(0)=0 \) and \( y^\prime(1)+y(1)=0 \). These conditions strongly influence the form of our solution and eliminate certain possibilities, guiding us towards the correct set of eigenfunctions.

Homogeneous ODE

A homogeneous ODE is one where the function and all its derivatives are proportional to the same function. In simpler terms, there's no extra term being added or subtracted from the equation, its right-hand side is zero. This was the starting point in our problem with \( y'' + 2y = 0 \). Solving homogeneous ODEs typically involves finding the general solution, which encompasses all possible solutions, characterized by constants that will be determined by boundary conditions.

Orthogonality of Eigenfunctions

Eigenfunctions associated with different eigenvalues of a Sturm-Liouville problem are orthogonal to each other under a certain inner product. Orthogonality is a critical feature since it allows us to use these functions as a basis to expand other functions, similarly to how vectors can form a basis in a vector space. Nevertheless, as we noticed in our solution progress, we cannot directly expand \( -x \) in terms of the eigenfunctions due to the orthogonality property not aligning with the given function shape. This challenge leads us to seek out alternative methods to progress the solution.

Method of Variation of Parameters

The method of variation of parameters is a technique to solve nonhomogeneous differential equations. It takes into account the form of the nonhomogeneous term and finds a particular solution by assuming parameters in the solution of the homogeneous equation are variable instead of constant. For our problem, once we've found the eigenfunctions, we could potentially employ this method to handle the inhomogeneous term \( -x \) and find a particular solution that fits our boundary conditions and complements the homogeneous solution.

Integrating Factor

An integrating factor is a function we multiply through an ordinary differential equation to make it easier to solve. It's particularly useful in solving linear first-order ODEs where direct integration isn't straightforward. For the more complex second-order inhomogeneous ODE in the exercise, an integrating factor can be utilized, though we would generally opt for variation of parameters or eigenfunction expansion, as it’s often simpler and more direct for these kinds of problems.