Assignment:
Consider the differential equation
y'' + p(t) y' + q(t) y = 0 t ∈ I,
where I = (a,b) and p,q are continuous functions on I.
(a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the equation.
(b) Let I = {-π, π}. Is {cos t, cos 2t} a fundamental set of solutions for the equation for some p(t),q(t)? If no, why not? If yes, what are the coefficient functions p(t) and q(t)?
Provide complete and step by step solution for the question and show calculations and use formulas.