I'm assuming full familiarity with the 3x3x3 solution. Faces are still denoted U,D,L,R,F,B. A lowercase letter means the slice that is inner to the uppercase letter. That is, "r" means the slice behind the right face.
We have 24 center pieces, 8 corner pieces and 12 edge pieces. I'll call two edge pieces that are adjacent a "wing".
As before, in a motion, if the letter is written as is, it means a clockwise turn. Otherwise, it's counter-clockwise. Also, usually I'll try to keep white face on top.
First, we will make sure that all centers are brought together correctly. Then, we will make sure that each wing consists of the same type edges. Then, we will apply the 3x3x3 solution (with some fixes).
Notice one thing - unlike the 3x3x3 cube - the white center pieces are not fixed to be accross the yellow center pieces. However, this does not mean that in the ultimate solution you may have yellow-white adjacent. In fact, seeing that there are no yellow-white corners, you definitely can't. You must make sure to get the orientation right. We'll see how it's done.
Getting the white center should be trivial. You should be able to form 2 adjacent white pieces in one of the faces F-R-B-L, position them vertically, and slide it up.
Now, do the same thing once more, without screwing up the top slice. I'll show as an example where there is a bottom slice center involved too.
Now that we have our first center, we'll make headway by getting the center accross - the yellow center. The action is the same - we'll need to get two centers in the F-L-B-R faces, line them up vertically, and shift them down. However, this will cause a problem - the white line will be shifted as well. But no problem - we'll set the yellow line aside, and return.
The last part might be a tad more complex. You need to move the white line "away". Let's repeat, this time assuming we already have one yellow line.
This point on we do only face moves and (Uu)/(Dd) shifts - never (Rr), (Ll), (Ff) or (Bb). This assures that the white/yellow centers aren't messed up. I usually utilize this trick - I try to make matching horizontal lines. For instance:
Here's a trick to swap any two strips:
This should be sufficient to do everything. First, make sure that green is across blue, then match up the rest. Now, if white is on top, the correct order is blue-orange-green-red (and not blue-red-green-orange). If you mixed it up, it's very easy to fix:
We want to orient the wings correctly. There are two ways to do so. The first is pretty intuitive:
It's pretty easy - you line up the edge that is set to be fixed (red/green in this case), fix it, but then you've ruined the centers. So you move the wing up to "safety", bring a "sacrifice wing" back (the one in cyan), and rotate the centers back in place. For this to work, you must have a "sacrifice wing" - a wing you don't mind misplacing. Except the last round, you'll always have one.
For the last round you apply the following:
If you did everything correctly, your cube should look pretty much like this:
And now you can solve it just like a 3x3. If you can't see that - go play solitaire. However, two problems may arrive once you reach the bottom layer, and I'll address each.
When you reach the bottom layer, you get an odd number of correctly oriented edges. It looks something like this, and the fixing method is given. It's pretty difficult, but you have to memorize it.
This happens after you did the second - after you make wings math up their colors (step 5 in the 3x3), you see that two corners match their places, and two don't. Step 6 of 3x3x3 won't fix the situation, you'll forever be stuck with two misplaced corners. What you do in this case, is flip the wings, with the following manouver.
Now you have wrong edge matchups. You fix it with the usual manouver (step 5), and then proceed with the last step.
You're done!