Qualitative Methods for First Order DEs

Section 2.2 Qualitative Methods for First Order DEs

As I discussed in Section 2.1, many DEs are very difficult or impossible to solve directly (using known functions). Implicit or qualitative methods try to say something about the solutions without actually finding the solutions. Even when I can exactly solve an equation, these methods are still very useful for interpreting the behaviour of the solutions.

Subsection 2.2.1 Autonomous DEs and Phase-Line Analysis

Consider a population model \(p(t)\) with its autonomous DE.

\begin{equation*} \frac{dp}{dt} = f(p) \end{equation*}

There is a lovely piece of qualitative analysis for autonomous equations called the phase line analysis. This is hopefully familiar from Calculus I, but I’ll review the whole theory here, since it is such a good first example of a qualitative method. Phase line analysis looks at the right side of the equation and asks: what values of \(p\) set the right side to zero? What does this mean? When the right side of the differential equation is 0, the left side is 0 as well. The left side is the growth rate, so that means the growth rate is zero. Therefore, the special value of \(p\) where the right side is zero is a value of the population where there is no growth.

Definition 2.2.1.

In an autonomous DE, a value of the function where the derivative vanishes is called a steady state. Note this is a value of the dependent variable, not the independent variable. In a population that depends on time, a steady state is a population count, not a specific time.

If the population is exactly at its steady state, it will not change; steady states are constant populations which do no grow or decline. (For population models, I can make the reasonable assumptions that \(p \geq 0\) and \(p=0\) is always a steady state. In other models that may include negative values, the steady state could be any real number.)

Once I have found the steady states, I can ask what happens between each steady state. Assuming that the DE is reasonable, the sign of the derivative will either be always positive or always negative between the steady states. When it is positive, there is positive growth rate and the population increases. When it is negative, there is negative growth rate and the population decreases.

Definition 2.2.2.

In an autonomous DE, the direction of growth negative or positive, called the trajectory of the popluation.

Steady states and trajectories give a remarkably robust understanding of the population.

If the popluation is at a steady state, it doesn’t change.
If the popluation is not at a steady state, I look at the trajectory.
If the trajectory is positive, the popluation grows either to the closest larger steady state or to infinity.
If the trajectory is neagative, the population declines either to the closest smaller steady state or to zero.

I can summarize this information is a phase line diagram. I take a ray representing \(p \geq 0\) and put dots on the ray for the steady states. In between, I draw arrows to show the trajectories. Its best to see the phase line diagrams through examples.

Example 2.2.3.

Figure 2.2.4. A Phase Line Diagram for \(\frac{dp}{dt} = p^2 - p\)

\begin{equation*} \frac{dp}{dt} = p^2 - p \end{equation*}

The right side is zero when \(p=0\) or \(p=1\text{,}\) so those are the steady states. When \(p \in (0,1)\) the derivative is negative, so the trajectory is decreasing. When \(p \in (1, \infty)\text{,}\) the derivative is positive, so the trajectory is increasing. Figure 2.2.4 shows the resulting phase-line.

Example 2.2.5.

Figure 2.2.6. Phase Line Diagram for \(\frac{dp}{dt} = p^3 - 7p^2 + 10p\)

\begin{equation*} \frac{dp}{dt} = p^3 -7p^2 + 10p \end{equation*}

The right side factors as \(p(p-2)(p-5)\text{,}\) so it is zero then \(p=0\text{,}\) \(p=2\) or \(p=5\text{.}\) Those are the steady states. When \(p \in (0,4)\) the derivative is positive, so the trajectory is increasing. When \(f \in (2,5)\text{,}\) the derivative is negative, so the trajectory is decreasing. When \(p \in (5,\infty)\text{,}\) the derivative is positive, so the trajectory is increasing. Figure 2.2.6 shows the phase line.

Example 2.2.7.

Figure 2.2.8. The Logistic Phase-Line Diagram

This example is a specific instance of a form known as the logistic equation.

\begin{equation*} \frac{dp}{dt} = 4p-p^2 \end{equation*}

The right side is zero when \(p=0\) or \(p=4\text{,}\) so those are the steady states. When \(p \in (0,4)\) the derivative is positive, so the trajectory is increasing. When \(p \in (4, \infty)\text{,}\) the derivative is negative, so the trajectory is decreasing. Figure 2.2.8 shows the resulting phase-line.

Figure 2.2.9. Exponential and Logistic Growth

The previous example was a logistic equation. The general form of the logistic equation is

\begin{equation*} \frac{dp}{dt} = cp(K - p) \end{equation*}

for some constants \(c\) and \(K\) with \(K \gt 0\text{.}\) The logistic equation leads to logistic growth. logistic growth, \(p = K\) is always a steady state and the starting values of the population always want to revert to this steady state. \(K\) is called the carrying capacity of the population. From below, the population wants to grow up to the carrying capacity. From above, the population wants to decay down to the carrying capacity. After exponential growth, logistic growth is the most commonly used model for populations. Figure 2.2.9 shows both exponential and logistic growth (where the carrying capacity for the logistic model is at \(p=6\text{.}\))

Subsection 2.2.2 Direction Fields

If a first order differential equation is not autonomous, then the phase-line is too simple a tool to capture the details. However, if I can solve for the derivative term in the DE, I can write a first order DE in this way, for some expression \(F\) in two variables.

\begin{equation*} \frac{dy}{dx} = F(x,y) \end{equation*}

This allows a very useful interpretation: the left side is the slope of a graph, the right side is a function on \(\RR^2\text{,}\) giving a value at every point in the plane. Together, these data determine a slope at every point in the plane.

Definition 2.2.10.

A determination of a slope at all point in a subset \(U\) of \(\RR^2\) is called a direction field or a slope field.

A direction field give a qualitative sense of the solutions of the differential equation, since those solutions are functions whose graphs must match the direction field. I can even draw curves that following the directions of the direction field. Such curves are called integral curves and they give a good picture of the expected solutions. This is best seen in examples. The first example is an autonomous equation, just to show that I can also apply this more complicated technique to autonomous equations. The examples following give direction fields and qualitative analysis for equations I already discussed in Subsection 2.1.4.

Example 2.2.11.

Figure 2.2.12. The Direction Field for \(\frac{dy}{dx} = y\text{.}\)

Figure 2.2.13. The Integral Curves for \(\frac{dy}{dx} = y\text{.}\)

\begin{equation*} \frac{dy}{dx} = y \end{equation*}

The slope at a point \((x,y)\) is \(y\text{,}\) so the slope at \((3,4)\) is \(4\text{,}\) the slope at \((-2,-3)\) is \(-3\) and the slope at \((32,0)\) is \(0\text{.}\) Figure 2.2.12 shows the direction field.

The solutions must fit the direction field. Therefore, if I can draw and understand the direction field, I get a sense of the solutions. Notice that since the direction field fills \(\RR^2\) (or a portion of it), I expect an infinite family of graphs of functions to match all the slopes. Figure 2.2.12 shows the graphs of the infinite family of solutions.

Example 2.2.14.

Figure 2.2.15. The Direction Field for \(\frac{dy}{dx} = x \sqrt{y}\text{.}\)

Figure 2.2.16. The Integral Curves for \(\frac{dy}{dx} = x \sqrt{y}\text{.}\)

\begin{equation*} \frac{dy}{dx} = x \sqrt{y} \end{equation*}

Figure 2.2.15 shows the direction field and the infinite family of solutions. I can see the entire family \(y = (\frac{x^2}{4} + c)^2\text{,}\) but I also see the singular solution \(y=0\text{.}\)

Example 2.2.17.

Figure 2.2.18. The Direction Field for \(\frac{dy}{dx} = \frac{4y}{x}\text{.}\)

Figure 2.2.19. The Integral Curves for \(\frac{dy}{dx} = \frac{4y}{x}\text{.}\)

\begin{equation*} \frac{dy}{dx} = \frac{4y}{x} \end{equation*}

There is a family of solutions \(y = cx^4\) which fits the direction field. The singular solutions are put together from one positive and one negative piece. Figure 2.2.18 shows the direction field and the infinite family of solutions.

Example 2.2.20.

Figure 2.2.21. The Direction Field for \(\frac{dy}{dx} = xy\text{.}\)

Figure 2.2.22. The Integral Curves for \(\frac{dy}{dx} = xy\text{.}\)

\begin{equation*} \frac{dy}{dx} = xy \end{equation*}

This is solved by \(y = ce^{x^2}\text{,}\) including \(c=0\) for \(y=0\) as a trivial solution. The direction field also shows the stability behaviour of the function: in this case, the functions grows very quickly away from the origin except for the stable and trivial \(y=0\) solution. Figure 2.2.21 shows the direction field and the infinite family of solutions.

Example 2.2.23.

Figure 2.2.24. The Direction Field for \(\frac{dy}{dx} = \frac{-x}{y}\text{.}\)

Figure 2.2.25. The Integral Curves for \(\frac{dy}{dx} = \frac{-x}{y}\text{.}\)

\begin{equation*} \frac{dy}{dx} = -\frac{x}{y} \end{equation*}

This is solved by \(y = \pm \sqrt{c-x^2}\text{,}\) which gives a series of circles. The direction field shows the bounded, relatively stable behaviour which is confirmed by the solutions. Notice that these solution have finite domains: I am not guaranteed solutions with domains that include all real numbers. Figure 2.2.24 shows the direction field and the infinite family of solutions.

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