# Singularity analysis of a nonlinear differential equation

I have a nonlinear differential equation of the form

$$y & # 39; & # 39; (x) + frac {y ((x)} {x} + Ce ^ {y (x)} = 0$$

Most of what I have read online and in textbooks deals only with the classification of singularities of second-order linear ODEs, nothing with nonlinear problems. Does anyone have a good reference to this?

I think one way to achieve that would be to linearize the system and analyze it $$x = 0$$, something like that:

$$begin {cases} y ((x) = w (x) \ w ((x) = – frac1xw (x) + Ce ^ {y (x)} end {cases}$$

and then find the Jakobian $$J$$ at the point $$z$$:

$$J (z) = begin {bmatrix} 0 & 1 \ Ce ^ {y (z)} & – frac1z end {bmatrix}$$

then analyze the equivalent equation $$y & # 39; & # 39; (x) + frac1zy ((x) + Ce ^ {y (z)} y (x) = 0$$, This obviously has a removable singularity $$z = 0$$but I do not know how that would relate to the original system.