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Math Help - PDE - find the general solution

  1. #1
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    PDE - find the general solution

    uxx-10uxt+21utt=0

    is hyperbolic

    reduce to canonical form to find the general solution

    -what is meant by canonical form
    i am using an old exam from another professor to study for a comp and we never talked about "canonical"
    or at least that i can find in my notes
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  2. #2
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    Quote Originally Posted by mlemilys View Post
    uxx-10uxt+21utt=0

    is hyperbolic

    reduce to canonical form to find the general solution

    -what is meant by canonical form
    i am using an old exam from another professor to study for a comp and we never talked about "canonical"
    or at least that i can find in my notes
    Canonical or standard form is the form for parabolic, hyperbolic and elliptic PDEs. There are as follows

    parabolic u_{xx} + \text{lots} = 0
    hyperbolic u_{tt} - u_{xx} + \text{lots} = 0
    modified hyperbolic u_{tx} + \text{lots} = 0
    elliptic  u_{tt} + u_{xx} + \text{lots} = 0

    where \text{lots} is lower order terms. Two go between the hyperbolic and modified hyperbolic introduce new coordinates r = t + x,\, s = t - x.

    Under the general change of variables

     <br />
r = r(x,y), s = s(x,y)<br />

    the first order derviatives transform as (the usual chain rule)

    u_t = u_r r_t + u_s s_t,\; u_x = u_r r_x + u_s s_x

    the second order derivatives transform as

     <br />
u_{tt} = r_t^2 u_{rr} + 2r_t s_t u_{rs} + s_t^2 u_{ss} + r_{tt} u_r + s_{tt} u_s<br />
     <br />
u_{tx} = r_t r_x u_{rr} + (r_t s_x + r_x s_t) u_{rs} + s_t s_x u_{ss} + r_{tx} u_r + s_{tx} u_s<br />
     <br />
u_{xx} = r_x^2 u_{rr} + 2r_x s_x u_{rs} +s_x^2 u_{ss} + r_{xx} u_r + s_{xx} u_s<br />

    If we substitute these three second order transforms into your PDE and target the modified form gives

    r_x^2 - 10 r_x r_t + 21 r_t^2 = 0,\;<br />
s_x^2 - 10 s_x s_t + 21 s_t^2 = 0<br />

    Now, both are the same equation and both factor

     <br />
(r_x - 3 r_t)(r_x - 7 r_t) = 0<br />

    Pick the first term for r and the second for s. These are first order PDEs and are easily solved giving

     <br />
r = R(3x + t),\;\; s = S(7x+t)<br />

    Now we'll pick easy r = 3x+t,\; s = 7x+t

    Under these change of variables you'll hit your modifed form. If you want regular form then choose

     <br />
r = (3x+t) + (7x+t) = 10 + 2t,\; s = (7x+t) - (3x+t) = 4x<br />

    If you wish you can scale each by 2.
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