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How to determine whether a point is a singular point of a curve
If I have a curve defined by a partial differential equation:
$$y’ = f(x,y)$$
Does that mean that the point in the plane $(x,y)$ where the derivative $f$ is zero is a singular point of the curve?
A:
No. A singular point of a curve is a point at which the derivative of the curve is zero, that is $\frac{d}{dt}y(t)\big|_{t=x} = 0$.
Your equation is not a differential equation. The thing with differential equations is that they are a formalization of a physical law, and thus be 100% accurate: you can check if the differential equation can describe any given curve.
However, any particular differential equation can be satisfied by one of many possible curves. This is the so-called isochronicity: while the differential equation gives you a curve, it does not tell you what a curve looks like. Only later on you can check the curve again to see if it satisfies the differential equation.
By the way, do not confuse this with the property of a singular point.
In your particular case, I believe you are talking about a property of curve: a singular point is a point at which the curve has infinite slope (i.e. it ends). However, that does not mean your equation should hold. Consider
$$
f(x,y)=y^2+x^3.
$$
The curve obtained from this equation ends at $(0,0)$, hence the first derivative is zero there. But in fact, it is not singular since the second derivative is not zero at $(0,0)$.
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