Using the characteristic method to solve
$$xu_x+uu_y=-tu, \quad\infty <x<\infty\;\; t>1\\ u(x,o)=x$$
here $\frac{dx}{x}=\frac{dt}{t}=-\frac{u}{t}$
Consider $\frac{dt}{t}=-\frac{u}{t}\implies t=-u+c_1$
now consider $\frac{dx}{x}=\frac{u}{-t}=\frac{du}{c_1-u}\implies x=c_2(c_1-u)$
Since $u(x,0)=x\implies x=s,t=0,u=s$
From I'm unable to eliminate the constants any help?
I always found it easier to write the characteristic equations in parametrized form: $$\frac{dx}{ds}=x,\quad x(0)=r$$ $$\frac{dy}{ds}=u,\quad y(0)=0$$ $$\frac{du}{ds}=-tu,\quad u(0)=r$$ This gives $x(s)=re^s, u(s)=re^{-ts}$ immediately, with the ODE for $y$ being $$\frac{dy}{ds}=re^{-ts}.$$ Once you solve this, you can find $s$ and $r$ in terms of $x$ and $y$, allowing you to express $u$ in terms of $x$ and $y$, which is the solution of the original PDE.