Finding the solution of d²y + 16y = 0 for y(x) with

Finding the solution of d²y + 16y = 0 for y(x) with the two boundary
dx²

conditions dy |_x=0 = 1 and dy |_x=π/2 = - 1 has
dx dx

1. no solution
2. exactly two solutions
3. exactly one solution
4. infinitely many solutions

(D² + 16)y = 0
A.E. → D² + 16 = 0
⇒ D = ± 4i
y = C₁cos 4x + C₂ sin 4x

	dy	= -4C₁sin 4x + 4C₂ cos 4x
	dx

	dy	\|_x=0 = -4C₁ × 0 + 4C₂ × 1 = 4C₂
	dx

4C₂ = 1
C₂ = 1/4
Now,

	dy	\|_x=π/2 = -4C₁ × sin4/2 + 4C₂ × cos = 4/2
	dx

	dy	\|_x=π/2 = 0 + 4C₂ = 4(1/2) = 1
	dx

But	dy	\|_x=π/2 = - 1 (Given)
	dx

Hence no solution.

Finding the solution of	d²y	+ 16y = 0 for y(x) with the two boundary
	dx²

conditions	dy	\|_x=0 = 1 and	dy	\|_x=π/2 = - 1 has
	dx		dx