Show that for all $\lambda \in \mathbb { R } , v _ { \lambda }$ can be expressed in the form:
$$v _ { \lambda } = \lambda w _ { 1 } + w _ { 2 }$$
with $w _ { 1 } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ the unique solution of the system
$$\left\{ \begin{array} { l }
- w _ { 1 } ^ { \prime \prime } ( x ) + c ( x ) w _ { 1 } ( x ) = 0 , x \in [ 0,1 ] \\
w _ { 1 } ( 0 ) = 0 \\
w _ { 1 } ^ { \prime } ( 0 ) = 1
\end{array} \right.$$
and $w _ { 2 }$ a function independent of $\lambda$ to be characterized.