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A singular nonlinear differential equationzσdwdz=aw+zwf(z,w),{z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w),whereσ&gt;1\sigma \gt 1, is considered in a neighbourhood of the pointz=0z=0located either in the complex planeC{\mathbb{C}}ifσ\sigmais a natural number, in a Riemann surface of a rational function ifσ\sigmais a rational number, or in the Riemann surface of logarithmic function ifσ\sigmais an irrational number. It is assumed thatw=w(z)w=w\left(z),a∈C⧹{0}a\in {\mathbb{C}}\setminus \left\{0\right\}, and that the functionffis analytic in a neighbourhood of the origin inC×C{\mathbb{C}}\times {\mathbb{C}}. Consideringσ\sigmato be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutionsw=w(z)w=w\left(z)in a domain that is part of a neighbourhood of the pointz=0z=0inC{\mathbb{C}}or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the propertylimz→0w(z)=0{\mathrm{lim}}_{z\to 0}w\left(z)=0is proved and an asymptotic behaviour ofw(z)w\left(z)is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point