grandes-ecoles 2018 Q1

grandes-ecoles · France · x-ens-maths__psi Second order differential equations Second-order ODE with initial or boundary value conditions

Let $\lambda \in \mathbb { R }$. Show that the problem
$$\left\{ \begin{array} { l } - v _ { \lambda } ^ { \prime \prime } ( x ) + c ( x ) v _ { \lambda } ( x ) = f ( x ) , x \in [ 0,1 ] \\ v _ { \lambda } ( 0 ) = 0 \\ v _ { \lambda } ^ { \prime } ( 0 ) = \lambda \end{array} \right.$$
admits a unique solution $v _ { \lambda } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$.

Let $\lambda \in \mathbb { R }$. Show that the problem

$$\left\{ \begin{array} { l } 
- v _ { \lambda } ^ { \prime \prime } ( x ) + c ( x ) v _ { \lambda } ( x ) = f ( x ) , x \in [ 0,1 ] \\
v _ { \lambda } ( 0 ) = 0 \\
v _ { \lambda } ^ { \prime } ( 0 ) = \lambda
\end{array} \right.$$

admits a unique solution $v _ { \lambda } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$.

Paper Questions

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