Results 1 to 3 of 3

Math Help - simultaneous equation problem

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    7

    simultaneous equation problem

    I have these two equations

    1 = 2e^{2r}cos\theta\frac{\partial r}{\partial x} -e^{2r}sin\theta\frac{\partial \theta}{\partial x}

    0 = 3e^{3r}sin\theta\frac{\partial r}{\partial x} +e^{3r}cos\theta\frac{\partial \theta}{\partial x}

    and I have to solve them simultaneously to obtain \frac{\partial r}{\partial x} and \frac{\partial \theta}{\partial x}. I hope this was the correct forum to post this question in.

    Any hints would be helpful, thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,492
    Thanks
    1391
    \displaystyle \left[\begin{matrix}2e^{2r}\cos{\theta} & -e^{2r}\sin{\theta}\\ 3e^{3r}\sin{\theta} & \phantom{-}e^{3r}\cos{\theta}\end{matrix}\right]\left[\begin{matrix}\frac{\partial r}{\partial x}\\ \frac{\partial \theta}{\partial x}\end{matrix}\right] = \left[\begin{matrix}1\\ 0\end{matrix}\right]

    \displaystyle  \left[\begin{matrix}2e^{2r}\cos{\theta} & -e^{2r}\sin{\theta}\\ 3e^{3r}\sin{\theta} & \phantom{-}e^{3r}\cos{\theta}\end{matrix}\right]^{-1} \left[\begin{matrix}2e^{2r}\cos{\theta} & -e^{2r}\sin{\theta}\\ 3e^{3r}\sin{\theta} & \phantom{-}e^{3r}\cos{\theta}\end{matrix}\right]\left[\begin{matrix}\frac{\partial r}{\partial x}\\ \frac{\partial \theta}{\partial x}\end{matrix}\right] =  \left[\begin{matrix}2e^{2r}\cos{\theta} & -e^{2r}\sin{\theta}\\ 3e^{3r}\sin{\theta} & \phantom{-}e^{3r}\cos{\theta}\end{matrix}\right]^{-1}\left[\begin{matrix}1\\ 0\end{matrix}\right]

    \displaystyle \mathbf{I}\left[\begin{matrix}\frac{\partial r}{\partial x}\\ \frac{\partial \theta}{\partial x}\end{matrix}\right] = \frac{1}{2e^{2r}\cos{\theta}\cdot e^{3r}\cos{\theta} - (-e^{2r}\sin{\theta})\cdot 3e^{3r}\sin{\theta}}\left[\begin{matrix}\phantom{-3}e^{3r}\cos{\theta} & \phantom{2}e^{2r}\sin{\theta}\\-3e^{3r}\sin{\theta} & 2e^{2r}\cos{\theta}\end{matrix}\right]\left[\begin{matrix}1\\ 0\end{matrix}\right]

    \displaystyle \left[\begin{matrix}\frac{\partial r}{\partial x}\\ \frac{\partial \theta}{\partial x}\end{matrix}\right] = \frac{1}{2e^{5r}\cos^2{\theta} + 3e^{5r}\sin^2{\theta}}\left[\begin{matrix}\phantom{-3}e^{3r}\cos{\theta}\\-3e^{3r}\sin{\theta}\end{matrix}\right]

    \displaystyle \left[\begin{matrix}\frac{\partial r}{\partial x}\\ \frac{\partial \theta}{\partial x}\end{matrix}\right] = \frac{1}{2e^{5r} + e^{5r}\sin^2{\theta}}\left[\begin{matrix}\phantom{-3}e^{3r}\cos{\theta}\\-3e^{3r}\sin{\theta}\end{matrix}\right]


    So \displaystyle \frac{\partial r}{\partial x} = \frac{e^{3r}\cos{\theta}}{2e^{5r}+ e^{5r}\sin^2{\theta}} and \displaystyle \frac{\partial \theta}{\partial x} = -\frac{3e^{3r}\sin{\theta}}{2e^{5r} + e^{5r}\sin^2{\theta}}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    7
    Thank you very much. This makes absolute sense.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simultaneous Exponential Equation Problem
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 1st 2011, 10:58 PM
  2. Simultaneous equation (?) word problem
    Posted in the Algebra Forum
    Replies: 5
    Last Post: November 2nd 2010, 10:48 AM
  3. Simultaneous equation word problem
    Posted in the Algebra Forum
    Replies: 5
    Last Post: September 27th 2009, 06:19 AM
  4. Simultaneous Equation Problem...
    Posted in the Algebra Forum
    Replies: 3
    Last Post: August 8th 2009, 07:12 AM
  5. Word Problem - Simultaneous Equation?
    Posted in the Algebra Forum
    Replies: 2
    Last Post: July 9th 2009, 06:13 AM

Search Tags


/mathhelpforum @mathhelpforum