Problem 5 - Arbeitsbuch Mathematik für Ingenieure

Question

Bestimmen Sie sämtliche reellen Eigenwerte \(\lambda\) und die zugehörigen Eigenfunktionen des Eigenwertproblems $$ \begin{aligned} &y^{\prime \prime}+\lambda \cdot y=0 \\ &y(0)-y^{\prime}(0)=y(\pi)-y^{\prime}(\pi)=0 \end{aligned} $$ Anleitung: Machen Sie die drei Fallunterscheidungen: \(\lambda>0, \lambda=0, \lambda<0\) und beachten Sie, dass die triviale Lösung \(y(x) \equiv 0\) keine Eigenfunktion ist.

Step-by-Step Solution

Answer

The real eigenvalues are \(\lambda_n = n^2\) for integer \(n\), with eigenfunctions \(y_n(x) = B \sin(nx)\).

Step 1: Explore the case \(\lambda = 0\)

Substitute \(\lambda = 0\) into the differential equation: \[ y'' = 0 \]Integrate twice to find the general solution: \[ y(x) = C_1 x + C_2 \]Apply the boundary conditions: \[ y(0) - y'(0) = C_2 = 0 \] \[ y(\pi) - y'(\pi) = C_1 \pi + C_2 - C_1 = C_1 (\pi - 1) = 0 \]The second boundary condition would lead to \(C_1 = 0\) if \(\pi eq 1\). However, this implies that \(y(x) = C_2\). Thus, only the trivial solution is possible for \(\lambda = 0\). Therefore, \(\lambda = 0\) does not yield any non-trivial eigenfunctions.

Step 2: Explore the case where \(\lambda > 0\)

Let \(\lambda = k^2\), where \(k > 0\). Substitute \(\lambda\) into the differential equation: \[ y'' + k^2 y = 0 \]The general solution is: \[ y(x) = A \cos(kx) + B \sin(kx) \]Apply the boundary conditions: \[ y(0) - y'(0) = A = 0 \] \[y(\pi) - y'(\pi) = A \cos(k\pi) + B \sin(k\pi) - (B k \cos(k\pi) - A k \sin(k\pi)) = 0 \]Since A = 0 from the first condition, this simplifies to \[ B \sin(k\pi) - B k \cos(k\pi) = B (\sin(k\pi) - k \cos(k\pi)) = 0 \] For non-trivial solutions, \(B eq 0\), leading to \[ \sin(k\pi) - k \cos(k\pi) = 0 \ Rightarrow k = 1, \sin(n\pi) where n is an integer \] Hence, possible values for \(\lambda\) are \(\lambda_n = n^2\) and corresponding eigenfunctions are: \[ y_n(x) = B \sin(nx) \]

Step 3: Explore the case where \(\lambda < 0\)

Let \(\lambda = -k^2\), where \(k > 0\). Substitute \(\lambda\) into the differential equation: \[ y'' - k^2 y = 0 \]The general solution is: \[ y(x) = A e^{kx} + B e^{-kx} \]Apply the boundary conditions: \[ y(0) - y'(0) = A + B - (Ak - Bk) = 0 \] \[ y(\pi) - y'(\pi) = A e^{k\pi} + B e^{-k\pi} - (Ak e^{k\pi} - Bk e^{-k\pi}) = 0 \]From these, set up the system of equations: \[ A (1 - k) + B (1 + k) = 0 \] \[ A e^{k\pi} (1 - k) + B e^{-k\pi} (1 + k) = 0 \] Solving these, for non-trivial \(A eq 0\) and \(B eq 0\), leads to a contradiction. Therefore, no non-trivial solutions exist for \(\lambda < 0\).

Key Concepts

Real Eigenvalues

Real eigenvalues are critical to solving differential equations involving eigenvalue problems. The eigenvalue \(\lambda\), which determines the behavior of the differential equation, can take on different forms: positive, zero, or negative. Each of these scenarios provides unique solutions and behaviors:

• \lambda > 0: Leads to sinusoidal solutions which can be represented with trigonometric functions like sine and cosine.
• \lambda = 0: Typically results in linear solutions that might not contribute any non-trivial solutions.
• \lambda < 0: Often leads to exponential-type solutions, which may not satisfy the boundary conditions.

Understanding these distinctions is vital in deriving the corresponding eigenfunctions and solving the differential equation accurately.

Eigenfunctions

Eigenfunctions correspond to specific eigenvalues and are essential functions that describe the system's behavior under given conditions. When we solve the differential equation: \[ y'' + \lambda \cdot y = 0 \] with various boundary conditions, we find the corresponding eigenfunctions:

• For \lambda = k^2 > 0: The general solution becomes a combination of sine and cosine functions, typically \ y(x) = A \cos(kx) + B \sin(kx) \ where A and B are constants determined by the boundary conditions.
• For \lambda = 0: The solution simplifies to linear functions \ y(x) = C_1 x + C_2, often yielding trivial solutions where no eigenfunctions exist.
• For \lambda = -k^2 < 0: The solution involves exponential functions \ y(x) = A e^{kx} + B e^{-kx}, but often leads to contradictions due to the boundary conditions.

Recognizing the form of eigenfunctions in various eigenvalue scenarios helps in solving complex differential equations.

Boundary Conditions

Boundary conditions are crucial constraints that the solutions must satisfy at specific points. For differential equation problems, they ensure the solution's uniqueness and relevance to the physical scenario:
In our problem, the boundary conditions are: \[ y(0) - y'(0) = 0 \ and \ y(\pi) - y'(\pi) = 0. \] These conditions limit the possible solutions and determine the constants within the general solution. For instance:

• For \lambda = k^2 > 0: \ These lead us to find specific values for constants A and B in the general solution to satisfy the conditions.
• For \lambda = 0: \ The boundary conditions imply that only the trivial solution (where \ y(x) = 0) is possible.
• For \lambda = -k^2 < 0: \ The boundary conditions often result in contradictions, leading to no non-trivial solutions.

By meticulously applying boundary conditions, we ensure the solutions are physically meaningful and mathematically correct.

Differential Equations Analysis

Analyzing differential equations, particularly those involving eigenvalue problems, requires a methodical approach:

• Identify the nature of the eigenvalue: Examine cases where \(\lambda\) is positive, zero, or negative. This influences the type of the general solution.
• Formulate the general solution: Solve the differential equation assuming different values for \(\lambda\). Each case will have its specific type of solutions.
• Apply the boundary conditions: Use given boundary conditions to find constants and specific forms of the solutions.
• Interpret the results: Analyze whether the solutions are trivial or non-trivial and if they fit the physical or theoretical context of the problem.

This systematic strategy aids in solving complex eigenvalue problems efficiently, ensuring the solutions' correctness and applicability.