Question 10.1.4: Solve the boundary value problem y′′ + y = 0, y(0) = 0, y(π)......

The general solution is given by equation (11),

y=c_{1}\cos x+c_{2}\sin x,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(11)

y=c_{1}\cos x+c_{2}\sin x,

and the first boundary condition requires that c_{1} = 0. Since sin π = 0, the second boundary condition is also satisfied when c_{1} = 0, regardless of the value of c_{2}. Thus the solution of problem (16) is y = c_{2}\sin x, where c_{2} remains arbitrary. This example illustrates that a homogeneous boundary value problem may have infinitely many solutions.