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.