?How to know if equilibrium points are stable or not. Is my solution correct

[cloud360]

Apologies, your derivative was correct. Your notation was not so good, though.
If y' means dy/dt, then y'' means d²y/dt², and that's not what is called for here. Instead of y'', you should use f^y or other notation to indicate that you're taking the partial with respect to y.

The object isn't merely to evaluate f(1, 2) and f^y(1, 2). You need to show that f(t, y) and f^y(t, y) are continuous in some rectangle around (1, 2). Can you do that?

Fy=-2(t^2-2)y+3y^2-1

i have no idea how to show this is continuous?

is there a method. if so i don't know what it is, cany you give me a link to a website which possibly teaches you how to know if somethig is continous.

i know how to prove a function is 1-1 though

 

[Mark44]

Both functions are defined at all point (t, y) in the plane. Since the only operations involved in their formulas are multiplication and addition, both functions are also continuous at every point (t, y) in the plane.

The basis ideas involved here are these: if g(x) and h(x) are continuous functions, the the sum (g + h)(x) and product gh(x) are continous. both f(t, y) and f_y(t, y) can be split up into the sum or product of simple linear or quadratic functions, which is enough to show that f(t, y) and f_y(t, y) are continuous. This is enough to show that the differential equation y' = f(t, y) has a unique solution that contains the point (1, 2). That's what you wanted to show in part e, so you're done.