I How can we identify non-linear singular differential equation

Summary
i am looking for singular , non-linear differetial equations but do not grasp a criteria to identify them.
i am doing research to make criteria by which i can identify easily linear and non-linear and also identify its singular or not by doing simple test.please help me in this regard.
 

Delta2

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I believe the most easy criteria to identify a non linear ODE is to look if it contains products of the unknown function with it self or with its derivatives.

For example the equation ##y^2-yy'+y''^3=x## is not linear for three reasons:
It contains product of the unknown function y with itself, that is the term ##y^2##
It also contains product of the unknown function y with its first derivative y', that is the term ##yy'##
It also contains product of the second derivative y'' with itself that is the term ##y''^3##
 

BvU

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And (especially in the beginning) there is always the simple test:
"Suppose ##y## statisfies the differential equation, does ##2y## satisfy it too ?"
 
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And (especially in the beginning) there is always the simple test:
"Suppose ##y## statisfies the differential equation, does ##2y## satisfy it too ?"
That's only for homogeneous differential equations (homogenouse in the sense of ##f(t, y, y', \dots) = 0##). For a simple example, consider y' - y = t
The general solution is ##y = ce^t + t - 1##, but ##2y = 2ce^t + 2t - 2## is not a solution to the DE.
 
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