QUESTION 3 You are asked to investigate beam buckling using different boundary conditions. For this problem, the governing differential equation is given by: d²y +2y=0 where μ² => and > > O dx2 a) Give one reason why boundary value problems (BVP) differ from initial value problems (IVP). [1 mark] b) Assuming the solution has an exponential form of y(x) as a function of μ. = ue, determine y [3 marks] c) Using Euler's formula etia = cos(ax)±i sin(ax), express the general solution y(x) as a summation of trigonometric functions. [3 marks] d) Determine the characteristic equations and non-trivial solutions yn for n = 1, 2, 3, and the following boundary conditions: i) y(0) = 0 and y(L) = 0 ii) y(0) = 0 and y' (L) = 0 [10 marks] e) Sketch the d) i) and ii) solutions for n = 2 over 0 < x < 1 and compare the deflection curves. Use L = 1. [Total marks: 20] [3 marks] Semester 2 2023-2024 KL5002: Further Mathematics Page 4 of 8