Proving Source, Sink, and Node of a 1st Order DE w/Taylor Series

[hkus10]

Challenge! Use Taylor series expansions to prove first-order Differential Equation

Suppose dy/dt = f(y) has an equilibrium point at y = y0 and
a) f'(y0) = 0, f''(y0) = 0, and f'''(y0) > 0: Is yo a source, a sink, or a node?
b) f'(y0) = 0, f''(y0) = 0, and f'''(y0) < 0: Is yo a source, a sink, or a node?
b) f'(y0) = 0 and f''(y0) > 0: Is yo a source, a sink, or a node?

Also, prove the answer you pick is true for each part!

I know that the answer for a) is source, b) sink, c) Node but I have no clue how to prove that is true.
Can anyone help me to start the question?

 

[Mark44]

Suppose dy/dt = f(y) has an equilibrium point at y = y0 and
a) f'(y0) = 0, f''(y0) = 0, and f'''(y0) > 0: Is yo a source, a sink, or a node?
b) f'(y0) = 0, f''(y0) = 0, and f'''(y0) < 0: Is yo a source, a sink, or a node?
b) f'(y0) = 0 and f''(y0) > 0: Is yo a source, a sink, or a node?

Also, prove the answer you pick is true for each part!

I know that the answer for a) is source, b) sink, c) Node but I have no clue how to prove that is true.
Can anyone help me to start the question?

How does your textbook define these terms: source, sink, node?

 

[hkus10]

Let use an example to illustrate source, sink, and node
For example, let assume the equilibrium points are y = -3 and y = 2. dy/dt < 0 fir -3 < y < 2, and dy/dt > 0 for y < -3 and y > 2. Given this information, y = -3 is a sink and y = 2 is a source.

Node just mean if the left hand and the right hand of an equilibrium has the same side of derivative.