5. Recall that a 1* order, autonomous equation can be written in general as = f(y). Consider the equation: - y3 – y? – 2y dr (a) Find the equilibrium solutions (critical points) of the equation. (b) Draw a phase diagram of y vs. f(y) (make sure to label the sketch). (c) Sketch of the actual solutions of r vs. y (make sure to label the sketch). (d) Characterize the stability of the equilibrium solutions (e.g. stable, unstable, semi-stable).

Differentiate the following expressions with respect to x: (i) y= (2x³-4) X+1 y = x-1 (ii) (iv) f(x)=(x³-3x)(2x²+3x+5) x²+2x y = x 3-x²-2x (v) y= x²-1 y y= √√x(√x+3) Complete Table 2.5 and then, find their derivatives. Table 2.5 (all) y = f(g(x)) (i) y= (6x-5)4 (ii) 1 y= (ii) y=√x²-1 u = g(x) y = f(u)

Are the functions f,g,f,g, and hh given below linearly independent? f(x)=0,    g(x)=cos(8x),    h(x)=sin(8x) If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer.

zwoled nwork dopas A. x-3 choupe D. x-6 B. X-4 C. x-5 10. The function y = f(x) is shown where a 0 and c> 0, then a potential equation might be: A. y=-(x-a)(x-b)(x-c)² B. y=(x-a)(x-b)(x − c)² C. y=-(x-a)(x-b)²(x − c) the a D. y=(x-a)(x-b)²(x-c) Record your answer in the numerical response section on your answer sheet.