I've been working on solving PDEs with conditions recently and have come across this question:
Obtain the explicit solution $u(x,t)$ to the IVP: $$u_t+uu_x=2t, \quad -\infty < x <\infty, \quad t>0,$$ $$u(x,0)=x, \quad -\infty<x<\infty.$$ Casting your answer as a function of $x$ and $t$ only.
I have had issues getting a solution in just terms of $x$ and $t$, and have been worried that I am misunderstanding how to tackle this. Any guidance is appreciated!
The comment by user Mattos is right. The method of characteristics for the quasi-linear case boils down to 2 coupled initial value ODEs
$$ \frac{du}{dt}~=~2t, \qquad u(t\!=\!0)~=~x_0,\tag{1a} $$
$$ \frac{dx}{dt}~=~u, \qquad x(t\!=\!0)~=~x_0, \tag{1b}$$
with solutions
$$ u~=~x_0+t^2,\tag{2a}$$
$$ x~=~(1+t)x_0 +\frac{t^3}{3}. \tag{2b}$$
Now eliminate $x_0$ from eq. (2a) by use of eq. (2b):
$$ u(x,t)~=~\frac{x-\frac{t^3}{3}}{1+t} +t^2. \tag{3}$$