You don't want a inductor, you want a electromagnet. Inductors are designed for their electrical properties with the external magnetic field being a byproduct. In fact, some inductors are designed to minimize the external electric field because this can cause interference and stray pickup in some circuits.
Electromagnets are designed to deliberately produce a external magnetic field. This is what you want because you want to interact with a magnetic guitar string.
I haven't seen a lot of deliberate electromagnets available as individual parts. Fortunately, these really are easy to make yourself. You wind some thin wire around a core. The more turns, the stronger the magnetic field for the same current, but also the higher the DC resistance.
For quick testing you can use a small iron or steel (not stainless steel) rod, like a nail. However, a conductive core also acts like the secondary of a transformer and essentially shorts out the transformer at AC. If you just want to make a controllable magnet that will be on long periods of time compared to the switch on time, then a iron core is fine. It will only add extra load when the magnetic field is changing. However, in your case you want to change the magnetic field at audio rates, so a conductive core is not a good idea.
What you want is a ferrite core. Ferrite is a material that does not conduct electricity but is still magnetic. Plain ferrite cores can be bought off the shelf in a variety of sizes. Fair-Rite is one company that comes to mind, and I know Mouser sells at least some of their stuff. A small ferrite rod is what you want.
Once you have the rod, wrap a few layers of tightly space magnet wire around it. This is around #30 wire with thin enamel insulation intended for exactly this kind of application.
To drive this, use a ordinary audio power amplifier intended for driving a loudspeaker. The impedance of your eletromagnet may be lower than the 8 Ω the power amp is expecting, so it might be a good idea to put a resistor in series with the coil, at least for starters to see how things go. A 4 Ω 2 W resistor should do it.
Since this is homework, I won't solve this completely for you, but I'll show you how to set up the equations:
First question "a". Let's name the bottom node "ground", and the top node "node 1", and call its voltage "v1". Now for each of the elements you can write a branch equation:
\$\dfrac{\mathrm{d}i_1}{\mathrm{d}t} = -\dfrac{v_1(t)}{3H}\$ (\$i_1\$ directed opposite of passive reference convention)
\$\dfrac{\mathrm{d}i_2}{\mathrm{d}t} = \dfrac{v_1(t)}{6H}\$
\$i_R(t) = \dfrac{v_1(t)}{2\Omega}\$
You also have a node equation for node 1:
\$i_1(t) = i_2(t) + i_R(t)\$
Given the initial conditions, you can also work out that the resistor current at t=0 is -1 A, and so \$v_1(0) = -1A \cdot 2 \Omega = -2V\$
From here you should be able to work out a solution for the different currents over time. Since you only have one storage element (the two inductors in parallel are equivalent to a single inductor with 6*3/(6+3) = 2 H inductance), you'll most likely end up with v1(t) decaying exponentially, and then being able to work out the individual inductor currents from there.
For equation (b), follow the same method: write all the independent branch and node equations you can, then combine and simplify until you have a solvable set of equations.
Best Answer
I'm a little surprised that the book tells you nothing about how to actually make the inductors. For AM, you generally make what's called a 'loopstick' antenna, like so:
The idea is that you wind a primary and a secondary coil such that the flux in one affects the flux in the other, and vice versa, hence it acts as a transformer.
Whenever you wind a coil, the inductance is proportional to the number of turns and to what's called the permittivity of the substance inside the coil. Ferrite has a much larger permittivity than air, which is why it's frequently used in inductors. See the Amateur Radio SE for a more detailed discussion of ferrite parameters. Of course, you can also make air coils, but you will need many more turns.
A resource like the ARRL handbook will give you rules of thumb for getting the correct inductance. Having an inductance meter (or a function generator and 'scope) helps for double-checking.