A function y(t) satisfies the differential equation

Question

A function y(t) satisfies the differential equation

dy/dx = y⁴ − 4y³ + 3y²

(a) What are the constant solutions of the equation? (Enter your answers as a comma-separated list.)

y= ??

(b) For what values of y is y increasing? (Enter your answer in interval notation.)

y=

(c) For what values of y is y decreasing? (Enter your answer in interval notation.)

y=

Answer 1

(a) Constant solutions, also called equilibrium solutions, satisfy dy/dx=0.

If we factor the right-hand side of the equation, we have

dy/dx = y²(y-1)(y-3). Setting the right-hand side equal to zero, we obtain the constant solutions y=0, y=1, y=3.

(b) Solutions are increasing when dy/dx > 0. Note that dy/dx > 0 for the set of y values (-∞,0) ∪ (0,1) ∪ (3,∞) Hence solutions are increasing when y in is the set (-∞,0) ∪ (0,1) ∪ (3,∞) .

(c) Solutions are decreasing when dy/dx < 0, and dy/dx < 0 when 1 < y < 3. That is, solutions are decreasing when y is in the set (1,3).