Solve the initial value problems for $u_t+2u_x=0$

Question

$u_t+2u_x=0$ initial value is $u(-1,x)=\frac{x}{1+x^2}$ Using the characteristic method i find that $\zeta= x-2t$ so the solution will be $$u(t,x)=\frac{x-2t}{1+(x-2t)^2}$$ so therefore when i plug in the initial value of $u(-1,x)$, i should of get $\frac{x}{1+x^2}$. So i thought the initial value was wrong, when i was talking to my teacher he told me there is a way to rewrite the solution so the initial value can satisfy. can anyone help me please

Answer 1

You have found the solution such that $u(0,x)=x/(1+x^2)$, but the one you want satisfies $u(-1,x)=x/(1+x^2)$. All you need is a translation in time.