For each of the phase line portraits drawn below write down

For each of the phase line portraits drawn below, write down a first order differential equation of the form x\' = fix). There are infinitely many possible answers for each portrait. The simplest approach is to use a polynomial for f(x).

Solution

EX 440. : x\'= (x+3)(x-3) has stable equilibrium solution at x₁= -3 and unstable at x₂= 3 as the direction field suggests. The values x₁ and x₂ are zeros of parabola f(x)= (x+3)(x-3). This is a parabola which is symmetric @ x-axis i.e f(x)

EX 441 : x\'= (x-3)(x+3) has stable equilibrium solution at x₁=3 and unstable at x₂=-3 as the direction field suggest. The values x₁ and x₂ are zeros of parabola f(x) = (x-3)(x+3). This is a parabola which is symmetric @ x-axis i.e f(x)