Solve the PDE. Use xi for the variable $\xi$.

PDE: $\quad u_{t} - 3 x u_{x} + 6 u = 0 , \quad -\infty < x < \infty$ and $t > 0$.
IC: $\quad u(x,0) = 8 \sin(x)$

First change to characteristic coordinates $(\xi,s)$ where $s=t$ and
$\xi = {}$
picked so that when $s=t=0$ then $\xi = x$, and that the equation becomes an ODE in $s$:
$u_s + {}$ ${} = 0$
with initial condition $u|_{s=0} = {}$

Solving this ODE, write $u$ in terms of $(\xi,s)$:
$u = {}$

Finally write $u$ in terms of $(x,t)$:
$u = {}$

help (formulas)