Is there any way, except trial and error, to find an isomorphism for these two graphs? 
How can I tell that these graphs are isomorphic and how can I show it? 

An easy technique for showing is to find the adjacency matrix of these graphs.If the adjacency matrix of one graph can be obtained from other by a suitable permutation of rows/columns then they are isomorphic otherwise not.
The adjacency matrix $A=(a_{ij}):=$
$a_{ij}=1$ if there exists an edge from $v_i$ to $v_j$
   =0 ,otherwise

