Changing coefficients: In looking at the equation zww" + ww' − zw'2 = i(1 + w4)/4, it struck me that (a) we should look at other solutions, and (b) see what happens if we change the coefficients on the right. But note that if we multiply either the independent or dependent variable by a constant, changing w(z) to u(z) = rw(sz), we don't materially affect its behavior. The factor s will rotate and/or dilate the picture, and r will permute the colors. Each such function (as well as 1/w) satisfies a similar differential equation, with different coefficients on the right hand side.
Therefore, if we look at the equation in the form zww" + ww' − zw'2 = a − bw4, by choosing the proper a and b, we can manipulate the solution so its initial value is 1, so restricting to this assumption loses no generality. The choice of the minus sign, by the way, was motivated by the fact that replacing w by 1/w simply switches a and b in this format. Here are graphs of solutions for a couple of choices.
If we let u = w2, we get a similar equation, but with multiples of u and u3 on the right. This is a Painlevé equation of type 3 (see page 345 of Ince's Ordinary Differential Equations or page 440 of Hille's Ordinary Differential Equations in the Complex Plane). Such an equation has a "Painlevé transcendent" as its solution.
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