Qualitative Analysis

Qualitative analysis helps show long term behaviour of a differential equation, as well as it’s potential equilibrium solutions without solving the entire equation.

Generally, the steps for qualitative analysis involve:

• Finding the equilibrium solutions of an ODE
• Sketching a phase plot
• Determining the stability of the equilibrium solutions

Once these steps are carried out, a family of solutions can be graphed.

Equilibrium Solutions §

An equilibrium solution is a constant solution of an ODE, implying that there is no change in the derivative. For first order ODEs: $\frac{dx}{dx}=f(x,t)$:

• $x(t)=c$, where $c$ is a constant
• $\frac{dx}{dt}=0$

Phase Plots §

A phase plot is a plot of $\frac{dx}{dt}$ as a function of $x$ (not $t$!) . Phase plots show both the equilibria solutions as well as the behaviour of the function near those solutions.

Phase plots are only useful if the ODE is autonomous. An autonomous ODE is one where the non-derivative function is only dependant on the independent variable:

$$\frac{dy}{dx}=f(x)isautonomous.\frac{dx}{dy}=f(x,y)isnot.$$

The stability of a solution can be seen via a phase plot.

• An equilibrium is stable if solutions near the equilibrium approach it. I.e. the limits of nearby solutions as $t\to \infty$ converge to the solution.
• An equilibrium is unstable is solutions near the equilibrium move away from it i.e. the limits of nearby solutions as $t\to \infty$ diverge from the solution.
• An equilibrium is semi stable if solutions from one side of the equilibrium move towards it, while those from the other side move away from it.

Examples §

1: Qualitative Analysis Of Simple Derivative

For the ODE $\frac{dx}{dt}=3x-1$, draw a phase plot and use it to sketch the family of solutions, including equilibria.

First, we need to find the equilibria solutions, which only occur when the derivative is zero and $x=c$: $\frac{dx}{dt}=0⟹3x-1=0⟹3c-1=0$ $c=\frac{1}{3}$ Using the phase plot, we can see that $\frac{dx}{dt}<0$ for x < 1/3 and $\frac{dx}{dt}>0$ for x > 1/3. Using this information to draw a family of solutions: