# Thread: PDE - find the general solution

1. ## 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

2. 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 $\displaystyle u_{xx} + \text{lots} = 0$
hyperbolic $\displaystyle u_{tt} - u_{xx} + \text{lots} = 0$
modified hyperbolic $\displaystyle u_{tx} + \text{lots} = 0$
elliptic $\displaystyle u_{tt} + u_{xx} + \text{lots} = 0$

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

Under the general change of variables

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

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

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

the second order derivatives transform as

$\displaystyle u_{tt} = r_t^2 u_{rr} + 2r_t s_t u_{rs} + s_t^2 u_{ss} + r_{tt} u_r + s_{tt} u_s$
$\displaystyle u_{tx} = r_t r_x u_{rr} + (r_t s_x + r_x s_t) u_{rs} + s_t s_x u_{ss} + r_{tx} u_r + s_{tx} u_s$
$\displaystyle u_{xx} = r_x^2 u_{rr} + 2r_x s_x u_{rs} +s_x^2 u_{ss} + r_{xx} u_r + s_{xx} u_s$

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

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

Now, both are the same equation and both factor

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

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

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

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

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

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

If you wish you can scale each by 2.