Chapter D1.1, Problem 50E

A second-order equation Consider the differential equation y”(t) – k²y(t) = 0, where k > 0 is a real number.

a.    Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁e^t + C₂e^–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) = C₁e^2t + C₂e^–2t.

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

d.    For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁ cosh kt + C₂ sinh kt, where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively.