In what follows we restrict to the case $n = 2$ from Part I. Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { 2 } \right) \right. \right.$ be a solution of equation (1). Show that there exist four real numbers $c _ { 1 } , c _ { 2 } , \varphi _ { 1 } , \varphi _ { 2 }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { 2 } c _ { i } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i } \right. \right.$$ (We recall that the two vectors $e _ { 1 } , e _ { 2 }$ are introduced in Question 4).
In what follows we restrict to the case $n = 2$ from Part I.
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { 2 } \right) \right. \right.$ be a solution of equation (1). Show that there exist four real numbers $c _ { 1 } , c _ { 2 } , \varphi _ { 1 } , \varphi _ { 2 }$ such that:
$$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { 2 } c _ { i } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i } \right. \right.$$
(We recall that the two vectors $e _ { 1 } , e _ { 2 }$ are introduced in Question 4).