Hello

I'm not sure here:

...

Are the following statements true or false? Prove your answer!

1)

The curve $\displaystyle t \mapsto (a*cos(t),b*sin(t),c*e^{-\frac{1}{2}*t}) $ is a solution of an initial value problem

2)

Let A be a 2x2 diagonal matrix. The initial value problem

$\displaystyle \dot{x}=Ax , x(0)=(a_1 , a_2 )$

has exactly one solution for each arbitrary $\displaystyle (a_1 , a_2 ) \in \mathbb{R}^2$

...

1)

Well we call this curve $\displaystyle \gamma$ then $\displaystyle \gamma(0)=(a,0,c)$ but is this already enough for being an initial value problem because

$\displaystyle \gamma'(t)=(-asin(t),bcos(t),\frac{-c}{2}*e^{\frac{-t}{2}})$ so the first two components of $\displaystyle \gamma'$ aren't a linear combination of any entry of $\displaystyle \gamma$??? How can I argue here?

2)

Here I think this is true because the solution for $\displaystyle \dot{x}=c*x$ in the one-dimensional case is unique according to our lecture. But how to prove this?

What do you think?

Regards