Chapter 4.4, Problem 41E

In Problems 41 and 42 solve the given initial-value problem in which the input function g(x) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that y and y′ are continuous at x = π/2 (Problem 41) and at x = π (Problem 42).]

41. y″ + 4y = g(x), y(0) = 1, y′(0) = 2, where $g(x)=\{\begin{array}{l}\sin x,0\le x\le \pi \text{/2}\\ 0,x>\pi \text{/2}\end{array}$