Fundamental Set of Solutions to an ODE
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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.
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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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