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Math Help - Change of coordinates question for a pde

  1. #1
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    Change of coordinates question for a pde

    Hello,

    I am struggling with the following question:

    Find all the solutions of the equation

    (d/dt (u(x,t))) -(d/dx (u(x,t))) +2u(x,t)=0

    Hint: a change on coordinates to simplify (d/dt (u(x,t))) -(d/dx (u(x,t)))

    Any help woyld be appreciated.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Change of coordinates question for a pde

    Quote Originally Posted by lordslytherin View Post
    Hello,

    I am struggling with the following question:

    Find all the solutions of the equation

    (d/dt (u(x,t))) -(d/dx (u(x,t))) +2u(x,t)=0

    Hint: a change on coordinates to simplify (d/dt (u(x,t))) -(d/dx (u(x,t)))

    Any help woyld be appreciated.
    Maybe this will help you. See my last post here:

    http://www.mathhelpforum.com/math-he...-182858-2.html
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  3. #3
    MHF Contributor
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    Re: Change of coordinates question for a pde

    Quote Originally Posted by Also sprach Zarathustra View Post
    Maybe this will help you. See my last post here:

    http://www.mathhelpforum.com/math-he...-182858-2.html
    I think you and the OP are talking about different problems.

    To the OP. If you introduce a change of variable from (t,x) to (r,s) then from the chain rule

    u_t = u_r r_t + u_s s_t
    u_x = u_r r_x + u_s s_x

    so your original problem becomes

    u_r r_t + u_s s_t - \left( u_r r_x + u_s s_x\right) + 2u = 0
    or

    \left(r_t- r_x\right)u_r +\left(s_t - s_x\right)u_s + 2 u = 0.

    If you can chose r and s such that

    r_t - r_x = 0 and s_t-s_x = 1, the your PDE becomes u_s + 2u = 0 - and ODE!

    Here's one choice
    Spoiler:

    r = t+x, s = t


    See how that goes.
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