What is current in RL-circuit?
I do not give full solutions to homeworks, but when t&lt;0, there was a steady state. All transients were dead. Consider the inductor behave as a zero ohm wire until the switch turns. Then the inductor current starts to change gradually towards the new steady state value.

Calculate the initial value from the preceding. Then calculate the new steady state value and let the difference between the initial value and final value die off with time constant L/R where R = 5 ohms. Your time constant is wrong. Your solving method of the differential equation needs repairing.