1. **State the problem:** We are given the differential equation $$y'' + \lambda y = 0$$ with boundary conditions $$y'(0) = 0$$ and $$y(1) = 0$$. We need to find the eigenvalues \(\lambda\), eigenfunctions \(y(x)\), and verify orthogonality. 2. **Solve the characteristic equation:** The characteristic equation for the ODE is $$r^2 + \lambda = 0$$. The roots are $$r = \pm i\sqrt{\lambda}$$. We consider three cases based on \(\lambda\): - Case 1: \(\lambda = 0\) - Case 2: \(\lambda > 0\) - Case 3: \(\lambda < 0\) 3. **Case 1: \(\lambda = 0\)** The equation becomes $$y'' = 0$$. General solution: $$y(x) = A + Bx$$. Apply boundary conditions: - $$y'(x) = B$$, so \(y'(0) = B = 0\) implies $$B = 0$$. - $$y(1) = A = 0$$ implies $$A = 0$$. Thus, trivial solution only. No nontrivial eigenvalue. 4. **Case 2: \(\lambda = \mu^2 > 0\), where \(\mu = \sqrt{\lambda} > 0\)** General solution: $$y(x) = A \cos(\mu x) + B \sin(\mu x)$$ Apply boundary conditions: - $$y'(x) = -A \mu \sin(\mu x) + B \mu \cos(\mu x)$$, - $$y'(0) = B \mu = 0 \Rightarrow B = 0$$ (since \(\mu \neq 0\)). Now $$y(x) = A \cos(\mu x)$$. Apply $$y(1) = 0 \Rightarrow A \cos(\mu) = 0$$. For nontrivial solution $$A \neq 0$$, so $$\cos(\mu) = 0$$. This yields $$\mu = \left(n - \frac{1}{2}\right) \pi, \quad n = 1, 2, 3, \dots$$ Hence, eigenvalues: $$\lambda_n = \mu_n^2 = \left(n - \frac{1}{2}\right)^2 \pi^2$$ Eigenfunctions: $$y_n(x) = A \cos\left(\left(n - \frac{1}{2}\right) \pi x\right)$$ Usually normalize or set \(A=1\). 5. **Case 3: \(\lambda = -\alpha^2 < 0\), where \(\alpha > 0\)** General solution: $$y(x) = A \cosh(\alpha x) + B \sinh(\alpha x)$$ Apply boundary conditions: - $$y'(x) = A \alpha \sinh(\alpha x) + B \alpha \cosh(\alpha x)$$ - $$y'(0) = B \alpha = 0 \Rightarrow B = 0$$ Now $$y(x) = A \cosh(\alpha x)$$. Apply $$y(1) = 0 \Rightarrow A \cosh(\alpha) = 0$$. Since $$\cosh(\alpha) > 0$$ for all \(\alpha\), the only solution is trivial $$A = 0$$. Thus no nontrivial solutions for negative eigenvalues. 6. **Orthogonality:** Eigenfunctions $$y_n(x) = \cos\left(\left(n - \frac{1}{2}\right) \pi x\right)$$ corresponding to distinct eigenvalues are orthogonal on \([0,1]\) with respect to the standard inner product: $$\int_0^1 y_m(x) y_n(x) dx = 0, \quad m \neq n.$$