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simple pde finding parametric representation with characteristic method

Hello,

I need to find parametric representation for part d.

$\displaystyle dr/dt + r*dr/dx = -xr$

Here is what I did but I keep not getting good answer

$\displaystyle dx/dt = r$

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

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

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

To find x I integrate from zero to t;

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

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

Can someone please please help me :(