# Continuous Functions, Fundamental Set of Solutions and Coefficient Functions

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Consider the attached differential equation 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 attached equation.

(b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solutions for the attached equation for some p(t),q(t)? If no, why not? If yes, what are the coefficient functions p(t) and q(t)?

NOTE: No computer, no calculator. Show how you would have done things by hand. Thanks so much!

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(a) For any linear ODE, the set of it's solutions is a vector space, that means any solution can be expressed as a linear combination of the basis.

As in any other vector space, a basis must contain only linear independent vectors

If the ODE is of degree (n), it will have (n) fundamental solutions (the basis), that means the dimension of the solutions vector space is (n)

In our example, the set of fundamental solutions will have 2 linear independent functions that ...

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