How do I write the supermesh equations in this circuit?
I'll show you an easy way using mesh analysis plus one node. First look at the node at the top of the right-hand side \$2\Omega\$ resistor:

$$i_1+2A=i_1+i_3$$

from which you get immediately \$i_3=2A\$. Then you need two more equations for \$i_1\$ and \$i_2\$. Use the outer mesh and note that the current through the left-hand \$2\Omega\$ resistor is \$i_1+i_2+2A\$:

$$(i_1+i_2+2A)\cdot 2\Omega + i_1\cdot 4\Omega+2A (=i_3)\cdot 2\Omega=10V\tag{1}$$

The second equation comes from the mesh in the lower left of the circuit:

$$(i_1+i_2+2A)\cdot 2\Omega+i_2\cdot 1\Omega=10V\tag{2}$$

Solving (1) and (2) for \$i_1\$ and \$i_2\$ gives the desired result.