How can we identify non-linear singular differential equation

[wasi-uz-zaman]

TL;DR 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]

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]

And (especially in the beginning) there is always the simple test:
"Suppose $y$ statisfies the differential equation, does $2y$ satisfy it too ?"

 

[Mark44]

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.