DE Challenge:

Does there exist a function f(y) such that the differential equation dy/dt = f(y):

i. has an infinite number of nodes, and such that f(y) satisfies the constraint that

f(y) + g(y) = k, for a continuous function g and some constant k?

ii. has multiple equilibrium, none of which is attracting?

iii. has many equilibrium solutions below the t axis (ie y < 0)?

iv. has not equilibrium solutions other than three nodes?

v. where f(t, y ) = Pi * t/y

vi. has a solution (and what is the solution) when f(y) = |y|?

vii. has two and only two equilibrium solutions, both of which are sinks (attractors)?

viii. has multiple equilibrium solutions all of which are attracting?

ix. Can one create, for each n in the positive integers, a differential equation

dy/dt = f(y) that has exactly n equilibrium solutions? …. That has infinitely many equilibrium solutions?