From what I can gather you've gone as far as you can without knowing the value of k.
If I were doing this question I'd leave the equation for L in terms of t and k
Problem: Find the solution of the differential equation that satisfies the given intial condition.
The intial condition: .
Well, I noticed its a separable equation,
The problem here, is the question did not identify the K!
so I assumed its a constant.
then I rewrite the D.E. as:
By integrating both sides and using integration by parts, I obtained:
Solving for L,I got:
, i.e. .
I stopped here!