# PDE - find the general solution

• May 11th 2009, 06:55 PM
mlemilys
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
• May 12th 2009, 04:36 AM
Jester
Quote:

Originally Posted by mlemilys
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

$
r = r(x,y), s = s(x,y)
$

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

$
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
$

$
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
$

$
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
$

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,\;
s_x^2 - 10 s_x s_t + 21 s_t^2 = 0
$

Now, both are the same equation and both factor

$
(r_x - 3 r_t)(r_x - 7 r_t) = 0
$

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

$
r = R(3x + t),\;\; s = S(7x+t)
$

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

$
r = (3x+t) + (7x+t) = 10 + 2t,\; s = (7x+t) - (3x+t) = 4x
$

If you wish you can scale each by 2.