For $y''+p(t)y'+q(t)y=0$, prove that if $y_1$ and $y_2$ are a fundamental set of solutions, then they cannot have a common point $t^*$ of inflection unless $p$ and $q$ vanish simultaneously there.
We know that if $y_1$ and $y_2$ are two solutions of the ODE, then their Wronskian $W[y_1,y_2](t)$ satisfies $W' + p(t)W = 0;$; lemma 1 [1]. From this we can assert that $p(t)$ has to vanish at $t^*$, since we have $W'(t^*) = 0$ there.
How can we make an implication on $q(t)$?
Context: this is problem 2.1.17 of Braun's Differential Equations and Their Applications. [1]
Best Answer
Since $t^*$ is a common point of inflection of $y_1(t),y_2(t)$, one has $$ y_1''(t^*)=y_2''(t^*)=0. $$ Putting $y_i$ into the equation and evaluating at $t=t^*$, one has $$ \left\{\begin{array}{ll}p(t^*)y_1'(t^*)+q(t^*)y_1(t^*)=0, \\ p(t^*)y_2'(t^*)+q(t^*)y_2(t^*)=0. \end{array}\right. $$ Now the determinant of coefficient matrix regarding to $p(t^*),q(t^*)$ is the Wronskian $W(y_1,y_2)(t^*)$ which is not zero since $y_1,y_2$ are fundamental solutions and hence one must have $p(t^*)=q(t^*)=0$.