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An Expression For the Velocity When Acceleration is Given in Terms of Velocity

Suppose the acceleration of a body is given in terms of the velocity. How can we find an expression for the velocity?
If  
\[a=2v^2\]
  and  
\[v=1\]
  when  
\[t=2\]
  (in the appropriate units), then we can write
\[a= \frac{dv}{dt}=2v^2\]

Now separate variables.
\[\frac{dv}{v^2}=v^{-2}dv=dt\]

Now integrate in the usual way.
\[\int^v_1 v^{-2}dv = \int^t_2 dt\]

\[[- \frac{1}{v} ]^v_1 = t-2\]

\[- \frac{1}{v} - ( \frac{1}{1} )=t-2 \]

Making  
\[v\]
  the subject gives  
\[v= \frac{1}{3-t}\]
.