Solving Inviscid Burgers' using Similarity I want to solve the inviscid Burgers' equation: 
$$\begin{equation}
\frac{\partial u}{\partial t} + u\frac{\partial u }{\partial x}  = 0
\end{equation}$$
I want to reduce the PDE to an ODE by saying that$\ u $ is a function of: 
$$\ u = u(x, t , u_o)$$
I expressed $\ u $ as a dimensionless quantity in terms of length (x), time(t) and the initial velocity (u0): 
$$\ [u] = [x^a t^b u_o^c]$$
Yielding the similarity variable $\eta $:
$$\eta = \frac{x}{tu_0}$$
I then defined the following function: 
$$u = u_0(\frac{x}{tu_0})^a = u_0F(\eta)$$
Subsequently carrying out the change of variables: 
$$\begin{equation}
\frac{\partial u}{\partial t} = F'(\eta)\frac{\partial \eta}{\partial t} = F'(\eta)(\frac{-x}{t^2})
\end{equation}$$
$$\begin{equation}
\frac{\partial u}{\partial x} = F'(\eta)\frac{\partial \eta}{\partial x} = F'(\eta)(\frac{1}{t})
\end{equation}$$
Plugging it back into the differential equation I end up with a strange ODE: 
$$\ F(\eta)F'(\eta) = \eta F'(\eta) $$
I think somewhere I made an error. If I need to show more steps please let me know. Any help would be appreciated. 
 A: Let us derive self-similar solutions by making the similarity Ansatz
$$
x = |t|^{\alpha} \xi
\qquad\text{and}\qquad
u = |t|^{\beta} U(\xi)\, .
$$
Under this assumption, we therefore have $\xi = x |t|^{-\alpha}$, and
\begin{aligned}
u_t(x,t) &= \beta t^{\beta-1} U(\xi) - \alpha t^{\beta-1} \xi U'(\xi) \\
u_x(x,t) &= t^{\beta-\alpha} U'(\xi)
\end{aligned}
for positive times, so that the PDE rewrites as
$$
\beta U(\xi) - \alpha \xi U'(\xi) + t^{\beta - \alpha + 1} U(\xi) U'(\xi) = 0 \, .
$$
The previous equation reduces to an ODE if $\beta = \alpha-1$, which is assumed from now on.
Analytical solutions can be obtained for various scaling parameters $\alpha$. Due to the problem's invariance by uniform dilatation of space and time $(x,t) \mapsto (k x,k t)$, a natural scaling is $\alpha = 1$. This scaling corresponds to $\xi = x/t$, and leads to the differential equation $(U(\xi)-\xi)\, U'(\xi) = 0$ in OP. Non-trivial solutions are $U(\xi) = \xi$, i.e. $u(x,t) = x/t$.
A self-similar solution of this form is called a rarefaction wave, a.k.a simple wave.
