Proving fundamental set of solutions DE

[Panphobia]

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.

 

[LCKurtz]

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.

 

[Panphobia]

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.

 

[LCKurtz]

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.