# Maxima and minima property of a differential equation having fundamental solution

Let $y_1 (x)$ and $y_2 (x)$ form a fundamental set of solutions to the differential equation $$\frac{\mathrm d^2 y}{\mathrm d x^2}+p(x)\frac{\mathrm dy}{\mathrm dx}+q(x)y=0,\quad a\leq x\leq b,$$ where $p(x)$ and $q(x)$ are continuous in $[a, b]$, and $x_0$ is a point in $(a, b)$. Then

1. Both $y_1 (x)$ and $y_2 (x)$ cannot have a local maximum at $x_0$.
2. Both $y_1 (x)$ and $y_2 (x)$ cannot have a local minimum at $x_0$.
3. $y_1 (x)$ cannot have a local maximum at $x_0$ and $y_2 (x)$ cannot have local minimum at $x_0$ simultaneously.
4. Both $y_1 (x)$ and $y_2 (x)$ cannot vanish at $x_0$ simultaneously.

Can anyone help me how to solve these type of problem. My thoughts are:

Here the Wronskian would be nonzero. Each case we get it as if any solution has local minima or maxima at a point then its derivative would be $0$ there, so all the options are correct. Am i right?

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You are correct, except that the derivative reasoning applies to 1,2,3. For 4, you still use the Wronskian, but the derivatives are irrelevant for this case. –  copper.hat Dec 19 '12 at 7:51