1. **Problem 1 (a): Verify Sturm-Liouville system** The differential equation is $$y'' + \lambda y = 0,$$ with boundary conditions $$y(0) = 0,\quad y(1) = 0.$$ This matches the form of a Sturm-Liouville problem $$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + (\lambda w(x) - q(x))y = 0,$$ where here $$p(x)=1, w(x)=1, q(x)=0.$$ The boundary conditions are homogeneous and linear, so the system is Sturm-Liouville. 2. **Problem 1 (b): Find eigenvalues and eigenfunctions** The equation is $$y'' + \lambda y = 0$$ with $$y(0)=0, y(1)=0.$$ Assume $$\lambda = \mu^2.$$ The general solution is $$y(x) = A\cos(\mu x) + B\sin(\mu x).$$ Apply $$y(0)=0$$: $$A=0,$$ so $$y(x) = B\sin(\mu x).$$ Apply $$y(1)=0$$: $$B\sin(\mu) = 0$$ and since $$B \neq 0,$$ we get $$\sin(\mu) = 0 \Rightarrow \mu = n\pi,$$ where $$n=1,2,3,\dots$$ Eigenvalues: $$\lambda_n = (n\pi)^2.$$ Eigenfunctions: $$y_n(x) = \sin(n\pi x).$$ 3. **Problem 1 (c): Orthogonality and orthonormal functions** Eigenfunctions satisfy orthogonality with respect to weight $$w(x)=1$$: $$\int_0^1 \sin(n\pi x)\sin(m\pi x)\,dx = 0, \text{ for } n \neq m.$$ For $$n=m,$$ $$\int_0^1 \sin^2(n\pi x)\, dx = \frac{1}{2}.$$ Orthonormal functions are obtained by normalizing: $$\phi_n(x) = \sqrt{2} \sin(n\pi x).$$ 4. **Problem 2 (a): $y(0)=0$, $y'(1)=0$** Equation and general solution as before: $$y(x) = A\cos(\mu x)+B\sin(\mu x).$$ Apply $$y(0)=0$$: $$A=0,$$ so $$y=B\sin(\mu x).$$ Apply $$y'(1)=0$$: $$y'(x) = B\mu \cos(\mu x) \Rightarrow y'(1) = B\mu \cos(\mu) = 0.$$ Since $$B \neq 0$$ and $$\mu \neq 0,$$ we get $$\cos(\mu) = 0,$$ thus $$\mu = \frac{\pi}{2} + n\pi, \quad n=0,1,2,\dots$$ Eigenvalues: $$\lambda_n = \left(\frac{\pi}{2} + n\pi\right)^2.$$ Eigenfunctions: $$y_n(x) = \sin\left(\left(\frac{\pi}{2} + n\pi\right) x\right).$$ 5. **Problem 2 (b): $y'(0)=0$, $y(1)=0$** General solution: $$y = A \cos(\mu x) + B \sin(\mu x).$$ Apply $$y'(0)=0$$: $$y'(x) = -A\mu \sin(\mu x) + B\mu \cos(\mu x).$$ At $$x=0$$: $$y'(0) = B\mu = 0 \Rightarrow B=0.$$ Apply $$y(1)=0$$: $$A \cos(\mu) = 0 \Rightarrow \cos(\mu) = 0,$$ so $$\mu = \frac{\pi}{2}+ n\pi, \quad n=0,1,2,\dots.$$ Eigenvalues: $$\lambda_n = \left(\frac{\pi}{2}+ n\pi\right)^2.$$ Eigenfunctions: $$y_n(x) = \cos\left(\left(\frac{\pi}{2} + n\pi\right) x\right).$$ 6. **Orthogonality and normalization for Problem 2** For eigenfunctions $$\sin\left(\left(\frac{\pi}{2} + n\pi\right) x\right)$$ and $$\cos\left(\left(\frac{\pi}{2} + n\pi\right) x\right),$$ orthogonality holds with weight 1 similar to Problem 1 over $$[0,1]$$. Normalization constants are evaluated as: $$\int_0^1 \sin^2(\alpha_n x) dx = \frac{1}{2}, \quad \int_0^1 \cos^2(\alpha_n x) dx = \frac{1}{2},$$ where $$\alpha_n = \frac{\pi}{2} + n\pi.$$ Hence orthonormal functions are: $$\phi_n(x) = \sqrt{2} \sin(\alpha_n x), \quad \psi_n(x) = \sqrt{2} \cos(\alpha_n x).$$