# First-order inhomogeneous PDE

1. Jan 27, 2009

### mcl4

1. The problem statement, all variables and given/known data
Assume ut+cux = xt, u(x,0) = f(x) for t>0. Find a formula for u(x,t) in terms of f, x, t, and c.

3. The attempt at a solution
I don't really follow what the professor is doing in class, and his office hours and the textbook weren't much more help, so the only thing I know about PDE's is what I've read online. That said:

$$\frac{du}{dr}$$ = $$\frac{dx}{dr}$$ux+$$\frac{dt}{dr}$$ut

$$\frac{dt}{dr}$$=1
t=r
$$\frac{dx}{dt}$$=c
x=ct+c'
x0=c'
x=ct+x0

$$\frac{du}{dr}$$=xt=ct2+x0t
$$\int$$du=$$\int$$(ct2+x0t)dr
u(x,t) = ct2r+x0tr+c''
u(x0,0) = 0+0+c'' = f(x0)
u(x,t) = ct2r+x0tr+f(x0)
= ct3+(x-ct)t2+f(x-ct)
= xt2+f(x-ct)

but when I calculate ut and ux and substitute into the original equation I do not get xt.

Any pointers would be much appreciated!