Conductivity from Boltzmznn Equation for a metal

[Abolfazl]

I'm trying to show that conductivity of a metal in uniform Electric field is:

$$
\sigma=\int \frac{d\textbf{k}}{4\pi^3}\left (- \frac{\partial f}{\partial \epsilon} \right )\textbf{v(k)u(k)}
$$
where u(k) is a solution to the integral equation
$$
\textbf{v(k)}=\int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}\textbf{[u(k)-u(k')]}
$$
$$
g(\textbf{k})=f(\textbf{k})+\delta g(\textbf{k})
$$
\delta g(k) is of order of E.
I also want to derive to linear order in E an integral equation obeyed by \delta g.
I wrote:
$$
-\frac{e}{\hbar}\frac{\partial f}{\partial E} \textbf{v.E}=\int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}[\delta\textbf{g(k)}-\delta\textbf{g(k')}]
$$
so what is the next step?
this is problem 4 of chapter 16 Ashcroft and Mermin. I don't have any idea for part b of the problem too. Any help is appreciated.

 

[Asim Riaz]

The next step is to take the left-hand side of the equation and solve for $\delta \textbf{g(k)}$. This can be done by rearranging the equation to isolate the $\delta \textbf{g(k)}$ terms:$$\delta \textbf{g(k)}=\frac{e}{\hbar}\frac{\partial f}{\partial E}\textbf{v.E} + \int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}\delta\textbf{g(k')}$$Once you have solved for $\delta \textbf{g(k)}$, you can then substitute it into the expression for the conductivity, which will give you an expression for the conductivity in terms of the electric field.For part b of the problem, you need to use the integral equation obeyed by $\delta \textbf{g(k)}$ to derive a linear equation obeyed by $\delta \textbf{g(k)}$. This can be done by substituting the integral equation into the expression for $\delta \textbf{g(k)}$:$$\delta \textbf{g(k)}=\frac{e}{\hbar}\frac{\partial f}{\partial E}\textbf{v.E} + \int \frac{d\textbf{k'}}{(2\pi)^3}W_{kk'}\left[\frac{e}{\hbar}\frac{\partial f}{\partial E}\textbf{v.E} + \int \frac{d\textbf{k''}}{(2\pi)^3}W_{k'k''}\delta\textbf{g(k'')}\right]$$This equation can then be rearranged to isolate the $\delta \textbf{g(k)}$ terms, giving you a linear equation for $\delta \textbf{g(k)}$.