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Proving fundamental set of solutions DE

  • Thread starter Panphobia
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  • #1
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Homework Statement


Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.


The Attempt at a Solution



I figured that if p(t0) is not 0 or q(t0) is not 0 then its not a fundamental set of solutions. So I have to show for the three cases
i) p(t0) =/= 0 q(t0) = 0
ii) p(t0) = 0 q(t0) =/= 0
ii) p(t0) =/= 0 q(t0) =/= 0
That the Wronskian is 0, but I don't know what to do to relate p to q in the wronskian.
 

Answers and Replies

  • #2
LCKurtz
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Homework Statement


Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.


The Attempt at a Solution



I figured that if p(t0) is not 0 or q(t0) is not 0 then its not a fundamental set of solutions.
How did you "figure" that? The equation ##y''- y=0## has fundamental solution pair ##\{e^x,e^{-x}\}## and ##q(x)=-1##.

So I have to show for the three cases
i) p(t0) =/= 0 q(t0) = 0
ii) p(t0) = 0 q(t0) =/= 0
ii) p(t0) =/= 0 q(t0) =/= 0
That the Wronskian is 0, but I don't know what to do to relate p to q in the wronskian.
That is nonsense. ##p## and ##q## are coefficients, not a solution pair and they have nothing directly to do with the Wronskian. And your argument doesn't even try to use the fact that your solutions have a common inflection point.

Hint: Think about the properties of the Wronskian and its derivative.
 
  • #3
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Those are the hints that my professor gave to me, so that is how I figured that. Why is it significant that y1 and y2 have an inflection point at t0? That just means the second derivative is 0. It is asking me to prove that if is is not the case that p(t0) = q(t0) = 0 then they are NOT a funcdamental set of solutions, and there are three ways that p(t0), q(t0) cannot be zero.
 
  • #4
LCKurtz
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I think you have misquoted or misunderstood the statement of the problem. What is true about your equation is that as long as ##p(x)## is not identically zero, then if two solutions have a common inflection point at ##t_0##, they cannot be a fundamental set, period. It doesn't matter about the zeroes of ##p## and ##q##. What I have just stated can be proved using my previous hint.
 

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