Basic Mathematics05 marksNumerical: phase line differential equationPhase lines and qualitative analysis
Draw a phase line for the equation fracdydx = (y + 1)(y - 2) and use it to sketch solutions to the equation.
Draw a phase line for the equation $\frac{dy}{dx} = (y + 1)(y - 2)$ and use it to sketch solutions to the equation. [5]
- Autonomous ODE: $\dfrac{dy}{dx} = (y+1)(y-2)$ Set $\dfrac{dy}{dx} = 0$: $$(y+1)(y-2) = 0 \implies y = -1 \quad \text{or} \quad y = 2$$ These are the equilibrium solutions (constant solutions). Region Test Point $(y+1)$ $(y-2)$ $\frac{dy}{dx}$ Motion ---------------------------------------------...