# Wronskian of second order DE?

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

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 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?

## 1 Answer

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

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