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Let $y_1, y_2$ be two solutions of $$y''(t)+ay'(t)+by(t)=0 ~~~~~\text{for}~~~ t\in \Bbb{R}~~~~~ \text{and}~~~ y(0)=0$$ where $a,b$ are real constants.

Let $W$ be the Wronskian of $y_1$ and $y_2$, then is $W=0$ on whole real line or $W=c$ for some positive constant $c$ or is it a nonconstant positive function, or there exist $t_1,t_2\in \Bbb{R}$ such that $$W(t_1)<0<W(t_2)~.$$

Now I know that if the solutions are linearly dependent then the wronskian is zero and if they are linearly independent then it is of the form $~Ae^{f(t)}~$ so cannot be constant unless $a=0$. I have no idea how can we say anything like, there exist $t_1,t_2\in \Bbb{R}$ such that $~W(t_1)<0<W(t_2)~$ is true or not but either it can be zero or nonconstant positive and either is possible as we are not given whether solutions are linearly independent or not.

I am sure $y(0)=0$ has to play some role here too, but cannot figure out what?

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$W(0)=y_1(0).y'_2(0)-y'_1(0).y_2(0)$

As $y_1(0)=0=y_2(0)$, so $W(0)=0$.Can you take it from here?

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