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Initial value problem-application (multivariable calculus)

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data
    This is a solution to a problem that was on a quiz, and I am confused about how to do it. Especially lines
    two (<0,1,1>=v(0)=<C1, 1+C2, C3> --> C1=0, C2=0, C3=1 and
    five (<1,0,0>=r(0)=<1+ K1, K2, K3> -->K1=0, K2=0, K3=0
    How do you do these steps? Can someone walk me through this process?
    upload_2015-12-9_20-42-35.png

    I'm studying for my final, and I KNOW that this will be one there.

    Thank you in advance!
     
  2. jcsd
  3. Dec 9, 2015 #2

    RUber

    User Avatar
    Homework Helper

    Since ##v(t) = \int -\cos t \hat i -\sin t \hat j dt, ## you get ##v(t) = -\sin t \hat i + \cos t \hat j + \vec C ##.
    Note that C can be any constant vector.
    Plug in t = 0 and compare with your v(0) term to solve for vector C.
    ##v(0) = -\sin 0 \hat i + \cos 0 \hat j + \vec C = 0 \hat i + 1 \hat j + \vec C = \hat j + \hat k ##
    This gives you ##\vec C = \hat k ##. Put this back into your equation for v(t) and you get ##v(t) = -\sin t \hat i + \cos t \hat j + \hat k ##.
    Next, you integrate velocity to get position.
    ##r(t) = \int -\sin t \hat i + \cos t \hat j + \hat k dt = \cos t \hat i + \sin t \hat j + t \hat k + \vec K. ##
    Where, again, vector K is any constant vector.
    As before, put in t = 0 and compare with initial position ## \hat i ## to solve for the constant vector K.
    ##r(0) = \hat i = \cos 0 \hat i + \sin 0 \hat j + 0 \hat k + \vec K =1 \hat i + 0 \hat j + 0 \hat k + \vec K . ##
    This shows you that vector K is the zero vector, so you can write r(t) as
    ##r(t) = \cos t \hat i + \sin t \hat j + t \hat k . ##
     
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