simple pde finding parametric representation with characteristic method

Hello,

I need to find parametric representation for part d.

$dr/dt + r*dr/dx = -xr$
Here is what I did but I keep not getting good answer

$dx/dt = r$
$dr/dx = -x*r = exp(-xt)$

since at t = 0; characteristic $x = ct + xo$ we have $x = xo$

$dx/dt = r(xo, 0)*exp(-xt)$

To find x I integrate from zero to t;

I get: $x = r(xo,0)*[(1/x)*exp(-xt)] + xo$

but here's the problem my solution is not $xo$ for t = 0

Can someone please please help me :(