Exercise

1.

Show that $e^{x},xe^{x},c_{1}e^{x}+c_{2}xe^{x}$ are solutions of the differential equation $y^{\prime\prime} -2y^{\prime} + y = 0$. For $y(0) = 1,\ y^{\prime}(0) = -1$, find $c_{1},c_{2}$.

2.

$\sin{x},\cos{x}$ and their linear combinations are solutions to the differential equation $y^{\prime\prime} + y = 0$. What can you say about the vector space spanned by $\sin{x},\cos{x}$.