1. **State the problem:** Solve the differential equation $$y'' + \lambda y = 0$$ with boundary conditions $$y'(0) = 0$$ and $$y(1) = 0$$. 2. **Write characteristic equation:** The differential equation is linear with constant coefficients. Assume solution of form $$y = e^{rx}$$. Characteristic equation: $$r^2 + \lambda = 0\implies r = \pm i\sqrt{\lambda}$$. 3. **Consider cases based on $$\lambda$$:** - If $$\lambda > 0$$, general solution is $$y = A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x)$$. - If $$\lambda = 0$$, simplifies to $$y''=0$$, solution is $$y = Ax + B$$. - If $$\lambda < 0$$, set $$\lambda = -\mu^2$$, general solution: $$y = Ae^{\mu x} + Be^{-\mu x}$$. 4. **Apply boundary conditions for $$\lambda > 0$$:** - First condition $$y'(0)=0$$ gives $$-A\sqrt{\lambda}\sin(0) + B\sqrt{\lambda}\cos(0) = B\sqrt{\lambda} = 0 \implies B=0$$. - The solution reduces to $$y = A\cos(\sqrt{\lambda}x)$$. - Second boundary condition $$y(1) = 0$$ implies $$A\cos(\sqrt{\lambda}) = 0$$. 5. Since $$A \neq 0$$ (non-trivial solution), require $$\cos(\sqrt{\lambda}) = 0$$. - $$\sqrt{\lambda} = \frac{\pi}{2} + n\pi, n=0,1,2,...$$. 6. **Eigenvalues:** $$\lambda_n = \left(\frac{\pi}{2} + n\pi\right)^2$$ for $$n = 0,1,2,...$$. 7. **Eigenfunctions:** Corresponding eigenfunctions are $$y_n = \cos\left(\left(\frac{\pi}{2} + n\pi\right)x\right)$$. 8. **Check other cases:** - For $$\lambda = 0$$, the solution and boundary conditions lead to trivial solution. - For $$\lambda < 0$$, exponential solutions cannot satisfy $$y'(0)=0$$ and $$y(1)=0$$ nontrivially. **Final answer:** Eigenvalues $$\lambda_n = \left(\frac{\pi}{2} + n\pi\right)^2$$ and eigenfunctions $$y_n = \cos\left(\left(\frac{\pi}{2} + n\pi\right)x\right)$$ for $$n=0,1,2,...$$.