1. ## Tedious PDEs PRoblem

I have 3 2nd order PDEs and I want to obtain 3 1st order PDEs from them. Following are the equations:

$
r_t +(rU)_x =tau([r_t+(rU)_x])_t + tau( (rU)_t + (rU^2 + p)_x )_x
$

$
(rU)_t + (rU^2 + p)_x = tau( (rU)_t + (rU^2 + p)_x )_t + tau( (rU^2 + p)_t + [U(rU^2 +3 p)]_x )_x
$

$
(rU^2 +(N+3) p)_t + [U(rU^2 +(N+5) p)]_x = tau( (rU^2 +(N+3) p)_t + [U(rU^2 +(N+5) p)]_x )_t
$

$
+ tau( [U(rU^2 +(N+5) p)]_t + [rU^4 + (N+8)pU^2 +(N+5) pRT]_x )_x
$

I want to obtain: $r_t,U_t,p_t$ or $r_t,U_t,T_t$

Can anybody give me pointers where should I start?

Thanks
Pipa

2. Is $\tau$ a constant? Also, is there a relation between $p\; \text{and}\; T$. What's N, and integer 0, 1 or 2? Also, is there a derivative missing in the second term, first equations?

3. Thanks for pointing that out. "Tau" is not a constant but can be treated like a constant for simplicity. The unit of Tau is time. N is just an integer and can be treated like a constant. P and T are related as follows:
$
p=rRT
$

P.S. I corrected the first equation above