How the equation that describes evolution of alkyl nitrates has been derived?
You have to quote a bit more of the paper for context, namely the reaction scheme. In the paper, it is given as 

$$\ce{RH -&gt;[\beta k_\mathrm{A}] RONO2 -&gt;[k_\mathrm{B}] P}$$

The first thing to notice is that the concentration of $\ce{RH}$ as a function of time is given by the first-order integrated rate law

$$[\ce{RH}]_t = [\ce{RH}]_0\exp(-\beta k_\mathrm{A}t)$$

and therefore equation (11) becomes

$$\frac{\mathrm{d}[\ce{RONO2}]}{\mathrm{d}t} = \beta k_\mathrm{A}[\ce{RH}]_0\exp(-\beta k_\mathrm{A}t)- k_\mathrm{B}[\ce{RONO2}]$$

This is a ordinary differential equation and there are many ways of solving them (this is hardly the place to discuss ODEs; consult a maths book if you are interested). In this case, the way I did it is to multiply throughout by $\exp(k_\mathrm{B}t)$. I am going to use $c$ to represent $[\ce{RONO2}]$ to simplify the equation a little bit:

$$\begin{align}
\frac{\mathrm{d}c}{\mathrm{d}t} &amp;= \beta k_\mathrm{A}[\ce{RH}]_0\exp(-\beta k_\mathrm{A}t)- k_\mathrm{B}c \\
\exp(k_\mathrm{B}t)\frac{\mathrm{d}c}{\mathrm{d}t} + \exp(k_\mathrm{B}t)k_\mathrm{B}c &amp;= \exp(k_\mathrm{B}t)\beta k_\mathrm{A}[\ce{RH}]_0\exp(-\beta k_\mathrm{A}t) \\
&amp;= \beta k_\mathrm{A}[\ce{RH}]_0 \exp[(k_\mathrm{B}-\beta k_\mathrm{A})t]
\end{align}$$

Now if you consider the derivative of $\exp(k_\mathrm{B}t)c$ with respect to $t$, we have (by the product rule, and bearing in mind that $c$ itself is a function of $t$)

$$\frac{\mathrm{d}}{\mathrm{d}t}[c\exp(k_\mathrm{B}t)] = \frac{\mathrm{d}c}{\mathrm{d}t}\exp(k_\mathrm{B}t) + ck_\mathrm{B}\exp(k_\mathrm{B}t)$$

which is the same as the left-hand side of the equation above. The thing we multiplied by throughout, $\exp(k_\mathrm{B}t)$, is called an integrating factor. So, we can now integrate both sides:

$$\begin{align}
\frac{\mathrm{d}[c\exp(k_\mathrm{B}t)]}{\mathrm{d}t} &amp;= \beta k_\mathrm{A}[\ce{RH}]_0 \exp[(k_\mathrm{B}-\beta k_\mathrm{A})t] \\
\int_0^t \frac{\mathrm{d}[c\exp(k_\mathrm{B}t)]}{\mathrm{d}t}\mathrm{d}t &amp;= \beta k_\mathrm{A}[\ce{RH}]_0 \int_0^t \exp[(k_\mathrm{B}-\beta k_\mathrm{A})t] \mathrm{d}t \\
\left[c\exp(k_\mathrm{B}t)\right]_{t=0}^{t} &amp;= \beta k_\mathrm{A}[\ce{RH}]_0 \left(\frac{\exp[(k_\mathrm{B}-\beta k_\mathrm{A})t] - 1}{k_\mathrm{B}-\beta k_\mathrm{A}}\right) \\
c_t\exp(k_\mathrm{B}t) - c_0 &amp;= \beta k_\mathrm{A}[\ce{RH}]_0 \left(\frac{\exp[(k_\mathrm{B}-\beta k_\mathrm{A})t] - 1}{k_\mathrm{B}-\beta k_\mathrm{A}}\right) \\
&amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} ([\ce{RH}]_0 \exp[(k_\mathrm{B}-\beta k_\mathrm{A})t] - [\ce{RH}]_0) \\
&amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} ([\ce{RH}]_t \exp(k_\mathrm{B}t) - [\ce{RH}]_t\exp(\beta k_\mathrm{A}t)) \\
c_t\exp(k_\mathrm{B}t) &amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} ([\ce{RH}]_t \exp(k_\mathrm{B}t) - [\ce{RH}]_t\exp(\beta k_\mathrm{A}t)) + c_0 \\
\frac{c_t}{[\ce{RH}]_t}\exp(k_\mathrm{B}t) &amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} (\exp(k_\mathrm{B}t) - \exp(\beta k_\mathrm{A}t)) + \frac{c_0}{[\ce{RH}]_t} \\
\frac{c_t}{[\ce{RH}]_t} &amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} (1 - \exp[(\beta k_\mathrm{A}-k_\mathrm{B})t]) + \frac{\exp(-k_\mathrm{B}t)c_0}{[\ce{RH}]_0\exp(-\beta k_\mathrm{A}t)} \\
&amp;= \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} (1 - \exp[(\beta k_\mathrm{A}-k_\mathrm{B})t]) + \frac{c_0}{[\ce{RH}]_0}\exp[(\beta k_\mathrm{A} - k_\mathrm{B})t] \\
\end{align}$$

I have shown quite a lot of the intermediate steps, if anything seems puzzling just work through it, it is really just algebraic manipulation.

Replacing $c$ with $[\ce{RONO2}]$ gets you to nearly your desired equation

$$\frac{[\ce{RONO2}]_t}{[\ce{RH}]_t} = \frac{\beta k_\mathrm{A}}{k_\mathrm{B} - \beta k_\mathrm{A}} (1 - \exp[(\beta k_\mathrm{A}-k_\mathrm{B})t]) + \frac{[\ce{RONO2}]_0}{[\ce{RH}]_0}\exp[(\beta k_\mathrm{A} - k_\mathrm{B})t]$$

The only difference in the paper is that $\beta$ was sometimes dropped (i.e. some of the $\beta k_\mathrm{A}$'s simply became $k_\mathrm{A}$). I glanced through the paper and can't find a rationale for it, but I will admit I did not look at it in great detail.