Chapter D1.1, Problem 51E

Another second-order equation Consider the differential equation y″(t) + k²y(t) = 0, where k is a positive real number.

a.    Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁ sin t + C₂ cos t. You may assume that this function is the general solution.

b.    Verify by substitution that when k = 2, the general solution of the equation is y(t) = C₁ sin 2t + C₂ cos 2t.

c.    Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.