Question

Solve the Sturm-Liouville problem $$ y^{\prime \prime}+\lambda y=0, \quad y(0)+\alpha y^{\prime}(0)=0, \quad y(\pi)+\alpha y^{\prime}(\pi)=0 $$ where \(\alpha \neq 0\)

Short answer

In summary, to solve the given Sturm-Liouville problem, we first propose a general solution to the second-order differential equation and then apply the boundary conditions to obtain an equation involving the eigenvalue parameter λ and the constant α. This transcendental equation can be solved numerically to obtain specific eigenvalues and eigenfunctions for the problem. The significance of these eigenvalues and eigenfunctions lies in understanding the characteristics of the differential equation and its solutions under the given boundary conditions.

Step-by-step solution

Solve the Second-Order Differential Equation

To solve the given second-order differential equation \(y^{\prime\prime}+\lambda y=0\), we propose a general solution of the form \(y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)\), where \(A\) and \(B\) are constants. The first derivative, \(y'(x)\), is given by: \(-A\sqrt{\lambda}\sin(\sqrt{\lambda}x)+B\sqrt{\lambda}\cos(\sqrt{\lambda}x)\) And the second derivative, \(y''(x)\), is given by: \(-A\lambda\cos(\sqrt{\lambda}x)-B\lambda\sin(\sqrt{\lambda}x)\) Now, substitute the general solution and its second derivative back into the differential equation: \(-A\lambda\cos(\sqrt{\lambda}x)-B\lambda\sin(\sqrt{\lambda}x) +\lambda(A\cos(\sqrt{\lambda}x) +B\sin(\sqrt{\lambda}x))=0\)

Apply the Boundary Conditions

We need to apply the following boundary conditions: \(y(0) + \alpha y'(0) = 0\) and \(y(\pi) + \alpha y'(\pi) = 0\). For the first boundary condition, plug in \(x=0\) into the general solution and its first derivative: \(y(0) = A\cos(0) + B\sin(0) = A\) \(y'(0) = -A\sqrt{\lambda}\sin(0)+B\sqrt{\lambda}\cos(0) = B\sqrt{\lambda}\) Now, apply the first boundary condition: \(A+\alpha B\sqrt{\lambda} = 0\). We can rearrange this equation to obtain \(B=-\frac{A}{\alpha\sqrt{\lambda}}\). For the second boundary condition, plug in \(x=\pi\) into the general solution and its first derivative: \(y(\pi) = A\cos(\sqrt{\lambda}\pi) - \frac{A}{\alpha\sqrt{\lambda}}\sin(\sqrt{\lambda}\pi)\) \(y'(\pi) = -A\sqrt{\lambda}\sin(\sqrt{\lambda}\pi)-\frac{A}{\alpha}\lambda\cos(\sqrt{\lambda}\pi)\) Now, apply the second boundary condition: \(A\cos(\sqrt{\lambda}\pi) - \frac{A}{\alpha\sqrt{\lambda}}\sin(\sqrt{\lambda}\pi) + \alpha\left(-A\sqrt{\lambda}\sin(\sqrt{\lambda}\pi)-\frac{A}{\alpha}\lambda\cos(\sqrt{\lambda}\pi)\right) = 0\).

Solve for the Specific Solution

Upon simplifying the second boundary condition equation, we get: \(A[\cos(\sqrt{\lambda}\pi) - \frac{1}{\alpha\sqrt{\lambda}}\sin(\sqrt{\lambda}\pi) - \alpha\sqrt{\lambda}\sin(\sqrt{\lambda}\pi) - \lambda\cos(\sqrt{\lambda}\pi)] = 0\) Since \(A\) cannot be equal to zero (otherwise the solution would be trivial), we must have: \(\cos(\sqrt{\lambda}\pi) + \frac{1}{\alpha\sqrt{\lambda}}\sin(\sqrt{\lambda}\pi) - \alpha\sqrt{\lambda}\sin(\sqrt{\lambda}\pi) - \lambda\cos(\sqrt{\lambda}\pi) = 0\) This is a transcendental equation that typically cannot be solved analytically for general \(\lambda\). However, the equation can be solved numerically for specific values of \(\lambda\) and \(\alpha\) to obtain the eigenvalues and the corresponding eigenfunctions for the Sturm-Liouville problem.

Key concepts

Second-Order Differential Equation

Understanding second-order differential equations is fundamental in various areas of mathematics and physics. In the context of the Sturm-Liouville problem, we are dealing with a specific type where the equation takes the form \(y^{\prime\prime} + \lambda y = 0\). The term \(y^{\prime\prime}\) represents the second derivative of the function \(y\) with respect to its independent variable, and \(\lambda\) is a parameter that, in this context, will later prove to be related to eigenvalues.

To solve such an equation, we look for solutions that consist of a combination of sine and cosine functions because these are the solutions to the homogeneous linear second-order differential equations. The general solution proposed, \(y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)\), uses coefficients \(A\) and \(B\) that will be determined by the initial or boundary conditions given in the problem.

Boundary Conditions

Boundary conditions are constraints necessary for the uniqueness of the solution of differential equations. In the given Sturm-Liouville problem, the boundary conditions are \(y(0) + \alpha y^\prime(0) = 0\) and \(y(\pi) + \alpha y^\prime(\pi) = 0\), where \(\alpha eq 0\). These conditions are applied at the endpoints of the interval \[0, \pi\] and play a crucial role in determining the specific values for the coefficients \(A\) and \(B\) in the general solution.

By substituting \(x = 0\) and \(x = \pi\) into the general solution and its derivative, as shown in the step-by-step solution, we obtain two equations involving \(A\), \(B\), and \(\lambda\). These equations are then solved to express one coefficient in terms of the other and ultimately yield relationships that will be used to find eigenvalues and eigenfunctions, which are the essence of the Sturm-Liouville problem.

Eigenvalues and Eigenfunctions

The concepts of eigenvalues and eigenfunctions arise in many areas of mathematics, particularly in solving linear operator problems like the Sturm-Liouville problem. Eigenvalues \(\lambda\) are special values for which the differential equation admits non-trivial solutions \(y(x)\), known as eigenfunctions, that satisfy the given boundary conditions.

In this problem, once we have applied the boundary conditions and manipulated the resulting equations, we encounter a transcendental equation. This equation cannot be solved explicitly for general \(\lambda\), but requires numerical methods for specific cases. Each \(\lambda\) that solves this transcendental equation is an eigenvalue, and the corresponding solution \(y(x)\) is the related eigenfunction. These pairs \(\{\lambda, y(x)\}\) characterize the modal shapes or natural vibration modes of the system described by the Sturm-Liouville equation.

Finding specific eigenvalues and eigenfunctions can be complex and often involves approximations or computational methods. The eigenfunctions that arise from these problems have orthogonality properties, which means they can form a basis to represent more complex functions that satisfy the same boundary conditions.