Fundamental set of solutions to an ode

Question

Assignment:

Suppose that p and q are continuous on some open interval I, and suppose that y1 and y2 are solutions of the ODE y'' + p(t)y' + q(t)y = 0 on I.

(a) Suppose that y₁, y₂ is a fundamental set of solutions. Prove that z1, z2, given by z1 = y1 + y2, z2 = y1 - y2, is also a fundamental set of solutions.

(b) Prove that if y₁ and y₂ achieve a maximum or a minimum at the same point in I, then they cannot form a fundamental set of solutions on this interval.

(c) Prove that if y₁ and y₂ form a fundamental set of solutions on I, then they cannot have a common inflection point in I, unless p and q are both 0 at this point.

(d) If 0 is in I, show that y(t) = t^3 cannot be a solution of the ODE on I.

Provide complete and step by step solution for the question and show calculations and use formulas.