# fundamental set of solutions for second order linear ODE

Show that if $$y_1(x_0) =y_2(x_0) = 0$$ then $$y_1$$ and $$y_2$$ cannot be the fundamental set of solutions for asecond order linear homogenous ODE on an interval $$I$$ with $$x_0 \in I$$.

How can one prove the above statement. I know by Wronskian that $$W(y_1,y_2)(x_0) = y_1(x_0)y_2'-y_1'y_2(x_0) = 0$$, but I don't have any idea what can be inferred from this equation.

There is some constant $$c$$ with $$y_2'(x_0)=cy_1'(x_0)$$. Use the uniqueness theorem to show that $$z=cy_1-y_2$$ must be the zero solution.
Or more compact, set $$z(x)=y_1(x)y_2'(x_0)−y_1'(x_0)y_2(x)$$ and show that it is the zero solution, proving linear dependence of $$y_1,y_2$$.
• For $z$ you have an IVP with obviously $z=0$ as one solution. Mar 11, 2019 at 19:18