I am trying to understand following proof. I understand the set up however can't make the connection with the Picard Lindelöf Theorem. Can you please help me with this?
Statement: The homogeneous boundary value problem $$ w'' + a_1 w' + a_2 w = 0, \ \ \ w(0)=0=w(1)=0$$ has at most one linearly independent solution. Where $a_1,a_2 \in C[0,1]$
Proof: Let $w_1,w_2$ be two solutions of that system. Then there exist constants $\lambda_1,\lambda_2$ s.t. $\lambda_1 w_1'(0) + \lambda_2 w_2'(0)=0 \ $ and $\ |\lambda_1|+|\lambda_2| > 0$. From the homogeneous boundary conditions we also have that $\lambda_1 w_1(0) + \lambda_2 w_2(0)=0$. Now the Picard Lindelöf uniqueness theorem for initial boundary value problems implies that $\lambda_1 w_1(t) + \lambda_2 w_2(t)=0 , \ \ \ \forall t \in [a,b]$. Hence the two solutions are lin.dep.