explain the problem---**Question Stem:** Q: Solve the Sturm-Liouville eigenvalue problem: **Differential Equation and Boundary Conditions:** d²y/dx² + λy = 0, y(0) = 0, y(π) = 0 **Problem Explanation/Steps:** Step 1: Identify the Type of Equation The equation is a second-order linear homogeneous differential equation with boundary conditions, which forms a Sturm-Liouville problem. The general Sturm-Liouville form is: d/dx [p(x) dy/dx] + [λw(x) - q(x)] y = 0 For this problem: * p(x) = 1 * q(x) = 0 * w(x) = 1 Step 2: General Solution of the ODE Given: $\frac{d^2y}{dx^2} + \lambda y = 0$ We solve the auxiliary (characteristic) equation: $r^2 + \lambda = 0 \Rightarrow r = \pm i\sqrt{\lambda}$ So the general solution is: $y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$ Step 3: Apply Boundary Conditions First Condition: $y(0) = 0 \Rightarrow A \cos(0) + B \sin(0) = A \cdot 1 + B \cdot 0 = 0 \Rightarrow A = 0$ So the simplified solution becomes: ```text y(x) = B sin(sqrt(lambda) x) Second Condition: y(pi) = 0 => B sin(sqrt(lambda) * pi) = 0 Since B != 0 (non-trivial solution), we must have: sin(sqrt(lambda) * pi) = 0 => sqrt(lambda) * pi = n pi => sqrt(lambda) = n Thus, lambda_n = n^2, n = 1, 2, 3, ... Step 4: Write Final Answers Eigenvalues: [Checkbox checked] Eigenvalues: lambda_n = n^2, n = 1, 2, 3, ... [Checkbox checked] Eigenfunctions: [Down arrow symbol] ``` Eigenfunctions: $y_n(x) = \sin(nx)$, $n = 1, 2, 3, \dots$ Bonus (If you want to write more for full 10 marks): Orthogonality of Eigenfunctions: The eigenfunctions $\{\sin(nx)\}$ form an orthogonal set on the interval $[0, \pi]$ with respect to the weight function $w(x) = 1$: $\int_{0}^{\pi} \sin(nx) \sin(mx)dx = \begin{cases} 0 & n \ne m \\ \frac{\pi}{2} & n = m \end{cases}$