Determine and draw the graph of the current \( i _{s}(t) \)

The work I have so far is basically using the relationship of voltage and current in an inductor
1. \( V=L \frac {di}{dt}
\frac {di}{dt} = \frac {V} {L}
\)

Now I found the current going through the inductor to be
\( i = 2V_{s}(t)+i_{s}(t) \)
Take the derivative of both sides
\( \frac {di}{dt} = 2V_{s}'(t)+i_{s}'(t) \)
Now substitute in what we found in equation 1 here
\( \frac {V} {L}=2V_{s}'(t)+i_{s}'(t) \)

Here V is \( V_{s}(t) \) and L = \( \frac{1}{2} \)

\( \frac {V_{s}(t)}{\frac{1}{2}} =2V_{s}'(t)+i_{s}'(t)
2V_{s}=2V_{s}'+i_{s}'(t) \)

And I'm pretty much stuck here not sure how I would get \( i_{s}(t) \) much less graph it...

 

Is the independent source a simple fixed DC value? If it is not, then one would need to know what the time dependent relationship for that source Vs(t) is.

 

What is the equivalent circuit of a voltage source in parallel with a current source?

 

Now, having said that, if you are looking for i_s(t) then you can't ignore the current source. You could ignore it and find the inductor current, since the inductor won't know where it is getting it's current from. Then you can use the now-known inductor current and the current source current to find i_s(t).