Final answer:
The differential equation y" + 4y = Sin x likely has an unbounded solution due to resonance, while y" + 4y = Cos 2x does not.
Step-by-step explanation:
The question involves determining whether the solutions to differential equations y" + 4y = Cos 2x and y" + 4y = Sin x are bounded or unbounded. An unbounded solution is one that increases without limit as x approaches infinity. For linear differential equations with constant coefficients, a solution is typically unbounded if the non-homogeneous term (on the right side of the equation) is a solution to the corresponding homogeneous equation (the equation with the right side set to zero). In the case of y" + 4y = Cos 2x, because Cos 2x is not a solution to the homogeneous equation y" + 4y = 0, we expect the particular solution will be bounded. However, for y" + 4y = Sin x, since the equation y" + 4y = 0 has solutions that include Sin x, the inhomogeneous equation is expected to have an unbounded solution due to resonance.
The equation y'' + 4y = cos 2x has an unbounded solution. This means that as x approaches positive or negative infinity, y also approaches positive or negative infinity, respectively. The cosine function has a maximum value of 1 and a minimum value of -1, so the amplitude of the solution will be unbounded. The equation y'' + 4y = sin x also has an unbounded solution because the sine function oscillates between -1 and 1, so the amplitude of the solution will also be unbounded.