For the differential equation y′′−4y=0, a student proposes the family of functions y=C1e2x+C2e−2x. Which of the following justifies that this family represents the general solution of the equation?
(a) Because the characteristic equation r2−4=0 has two real roots, r=2 and r=−2, no arbitrary constants are needed.
(b) Because the functions e2x and e−2x are solutions, a single constant multiple already covers every possibility.
(c) Two independent constants appear, matching the equation’s second order.
(d) Because initial conditions can always be chosen to eliminate one constant, only one arbitrary constant is required.