First order RL circuit analysis problem
The current calculation with the open switch at \$t_0^-\$ is correct.

When the switch closes at \$t_0^+\$, you will have three current sources, the original \$5~A\$ that can be ignored, as well as \$i_1 = 2.4~A\$ and \$i_2 = 0.6~A\$.  Each current decay can hence be calculated taking the resistances into consideration.