How would one tell if the following ODE has any singular solutions?

Question

The ODE is $y'(x) + \frac{y(x)}{x}=-x^4y(x)^3$.

I found the solution to be:

$$y(x) = \frac{1}{x\sqrt{\frac{2}{3}x^3+C}}$$

but I'm not sure what is meant by "singular" solutions.

Thanks!

Answer 1

It is a solution to the differential equation that isn't actually included in the solution you found above if such a solution exists. For example, your solution says y cannot be 0. But y can be 0. Since 0+0=0

The singular solution is $y=0$