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Math Help - true or false: statement concerning initial value problems

  1. #1
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    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 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
    \dot{x}=Ax ,    x(0)=(a_1 , a_2 )
    has exactly one solution for each arbitrary (a_1 , a_2 ) \in \mathbb{R}^2
    ...

    1)
    Well we call this curve \gamma then \gamma(0)=(a,0,c) but is this already enough for being an initial value problem because

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


    2)
    Here I think this is true because the solution for \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
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  2. #2
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    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 A will just multiply the two unknown functions in the vector x(t)=(x_1(t),x_2(t)) by the constants on the diagonal. You then end up with two initial value problems c_1x_1(t)=x'_1(t), x_1(0)=a_1 and 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.
    Thanks from HallsofIvy
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  3. #3
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    Re: true or false: statement concerning initial value problems

    Thanks now it's all clear

    Regards
    Last edited by huberscher; November 21st 2012 at 06:20 AM.
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