Find the characteristic polynomial show that the

Question

y''(t)- α²y = 0 (α > 0):

(a) Find the characteristic polynomial.

(b) Show that the characteristic roots are ±β.

(c) Give the complete solution in terms of (real-valued) functions.

(d) Find a standard basis y₁(t), y₂(t) at t = 0 in terms of exponentials and convert to hyperbolic functions.