true or false: statement concerning initial value problems

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

Re: true or false: statement concerning initial value problems

For the the first statement just set up the initial value problem for which the given function is the solution. So you present the derivative and initial value you've found and show the function you were given solves this initial value problem.

The second is true two. A diagonal matrix $\displaystyle A$ will just multiply the two unknown functions in the vector $\displaystyle x(t)=(x_1(t),x_2(t))$ by the constants on the diagonal. You then end up with two initial value problems $\displaystyle c_1x_1(t)=x'_1(t), x_1(0)=a_1$ and $\displaystyle c_2x_2(t)=x'_2(t), x_2(0)=a_2$ each of which has a unique solution, which you'll have to show.

Re: true or false: statement concerning initial value problems

Thanks now it's all clear :)

Regards