# Fundamental Set of Solutions to an ODE

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.

https://brainmass.com/math/calculus-and-analysis/fundamental-set-solutions-ode-170921

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

When the two solutions are linearly dependent we have for some constant c:

Then:

So a condition for independence is that the Wronskian W(x) is non zero:

In our case we can write:

Therefore:

Thus form an independent fundamental solutions set as well.

A little theoretical detour:

The equation is:

The two ...

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A fundamental set of solutions for an ODE is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.