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kingwinner

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**1) Assume that p and q are continuous on some open interval I, and that y**

a) Prove that if {y

b) If 0 E I, prove that y(t)=t

_{1}and y_{2}are solutions of y'' + p(t)y' + q(t)y = 0 on I.a) Prove that if {y

_{1}, y_{2}} form a fundamental set of solutions on I, then they can't have a common inflection point in I, unless p and q are both 0 at this point.b) If 0 E I, prove that y(t)=t

^{3}cannot be a solution of the differential equation on I.=====================

1a) My first question: "A unless B", is this equivalent to "A if and only if not B". That is, do I have to prove

*both*directions for question 1a simply by seeing the word "unless"?

Now my thoughts about this problem:

Inflection point at c => f ''(c)=0 or f ''(c) does not exist.

But in this case, since we're given that y

_{1}and y

_{2}are solutions of the differential equation, this means that y

_{1}and y

_{2}must be twice differentiable, so we can reject the possiblity f ''(c) does not exist.

Now we have: inflection point at c => f ''(c)=0

Fundamental set of solutions means Wronskian is nonzero.

But now I am having a lot of trouble deriving one from the another...

1b) No clue...need some hints...

Thanks for your help!

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